
hoelzl@57235: (*  Title:      HOL/Probability/Distributions.thy
hoelzl@57235:     Author:     Sudeep Kanav, TU München
hoelzl@57254:     Author:     Johannes Hölzl, TU München
hoelzl@57254:     Author:     Jeremy Avigad, CMU *)
hoelzl@57235: 
hoelzl@57235: header {* Properties of Various Distributions *}
hoelzl@57235: 
hoelzl@50419: theory Distributions
hoelzl@57252:   imports Convolution Information
hoelzl@50419: begin
hoelzl@50419: 
hoelzl@57252: lemma (in prob_space) distributed_affine:
hoelzl@57252:   fixes f :: "real \<Rightarrow> ereal"
hoelzl@57252:   assumes f: "distributed M lborel X f"
hoelzl@57252:   assumes c: "c \<noteq> 0"
hoelzl@57252:   shows "distributed M lborel (\<lambda>x. t + c * X x) (\<lambda>x. f ((x - t) / c) / \<bar>c\<bar>)"
hoelzl@57252:   unfolding distributed_def
hoelzl@57252: proof safe
hoelzl@57252:   have [measurable]: "f \<in> borel_measurable borel"
hoelzl@57252:     using f by (simp add: distributed_def)
hoelzl@57252:   have [measurable]: "X \<in> borel_measurable M"
hoelzl@57252:     using f by (simp add: distributed_def)
hoelzl@57252: 
hoelzl@57252:   show "(\<lambda>x. f ((x - t) / c) / \<bar>c\<bar>) \<in> borel_measurable lborel"
hoelzl@57252:     by simp
hoelzl@57252:   show "random_variable lborel (\<lambda>x. t + c * X x)"
hoelzl@57252:     by simp
hoelzl@57252:   
hoelzl@57252:   have "AE x in lborel. 0 \<le> f x"
hoelzl@57252:     using f by (simp add: distributed_def)
hoelzl@57252:   from AE_borel_affine[OF _ _ this, where c="1/c" and t="- t / c"] c
hoelzl@57252:   show "AE x in lborel. 0 \<le> f ((x - t) / c) / ereal \<bar>c\<bar>"
hoelzl@57252:     by (auto simp add: field_simps)
hoelzl@57252: 
hoelzl@57252:   have eq: "\<And>x. ereal \<bar>c\<bar> * (f x / ereal \<bar>c\<bar>) = f x"
haftmann@57514:     using c by (simp add: divide_ereal_def ac_simps one_ereal_def[symmetric])
hoelzl@57252:     
hoelzl@57252:   have "density lborel f = distr M lborel X"
hoelzl@57252:     using f by (simp add: distributed_def)
hoelzl@57252:   with c show "distr M lborel (\<lambda>x. t + c * X x) = density lborel (\<lambda>x. f ((x - t) / c) / ereal \<bar>c\<bar>)"
hoelzl@57252:     by (subst (2) lborel_real_affine[where c="c" and t="t"])
hoelzl@57252:        (simp_all add: density_density_eq density_distr distr_distr field_simps eq cong: distr_cong)
hoelzl@57252: qed
hoelzl@57252: 
hoelzl@57252: lemma (in prob_space) distributed_affineI:
hoelzl@57252:   fixes f :: "real \<Rightarrow> ereal"
hoelzl@57252:   assumes f: "distributed M lborel (\<lambda>x. (X x - t) / c) (\<lambda>x. \<bar>c\<bar> * f (x * c + t))"
hoelzl@57252:   assumes c: "c \<noteq> 0"
hoelzl@57252:   shows "distributed M lborel X f"
hoelzl@57252: proof -
hoelzl@57252:   have eq: "\<And>x. f x * ereal \<bar>c\<bar> / ereal \<bar>c\<bar> = f x"
haftmann@57514:     using c by (simp add: divide_ereal_def ac_simps one_ereal_def[symmetric])
hoelzl@57252: 
hoelzl@57252:   show ?thesis
hoelzl@57252:     using distributed_affine[OF f c, where t=t] c
hoelzl@57252:     by (simp add: field_simps eq)
hoelzl@57252: qed
hoelzl@57252: 
hoelzl@57235: lemma (in prob_space) distributed_AE2:
hoelzl@57235:   assumes [measurable]: "distributed M N X f" "Measurable.pred N P"
hoelzl@57235:   shows "(AE x in M. P (X x)) \<longleftrightarrow> (AE x in N. 0 < f x \<longrightarrow> P x)"
hoelzl@57235: proof -
hoelzl@57235:   have "(AE x in M. P (X x)) \<longleftrightarrow> (AE x in distr M N X. P x)"
hoelzl@57235:     by (simp add: AE_distr_iff)
hoelzl@57235:   also have "\<dots> \<longleftrightarrow> (AE x in density N f. P x)"
hoelzl@57235:     unfolding distributed_distr_eq_density[OF assms(1)] ..
hoelzl@57235:   also have "\<dots> \<longleftrightarrow>  (AE x in N. 0 < f x \<longrightarrow> P x)"
hoelzl@57235:     by (rule AE_density) simp
hoelzl@57235:   finally show ?thesis .
hoelzl@57235: qed
hoelzl@57235: 
hoelzl@57235: subsection {* Erlang *}
hoelzl@57235: 
hoelzl@57235: lemma nn_intergal_power_times_exp_Icc:
hoelzl@57235:   assumes [arith]: "0 \<le> a"
hoelzl@57235:   shows "(\<integral>\<^sup>+x. ereal (x^k * exp (-x)) * indicator {0 .. a} x \<partial>lborel) =
hoelzl@57235:     (1 - (\<Sum>n\<le>k. (a^n * exp (-a)) / fact n)) * fact k" (is "?I = _")
hoelzl@57235: proof -
hoelzl@57235:   let ?f = "\<lambda>k x. x^k * exp (-x) / fact k"
hoelzl@57235:   let ?F = "\<lambda>k x. - (\<Sum>n\<le>k. (x^n * exp (-x)) / fact n)"
hoelzl@57235: 
hoelzl@57235:   have "?I * (inverse (fact k)) = 
hoelzl@57235:       (\<integral>\<^sup>+x. ereal (x^k * exp (-x)) * indicator {0 .. a} x * (inverse (fact k)) \<partial>lborel)"
hoelzl@57235:     by (intro nn_integral_multc[symmetric]) auto
hoelzl@57235:   also have "\<dots> = (\<integral>\<^sup>+x. ereal (?f k x) * indicator {0 .. a} x \<partial>lborel)"
hoelzl@57235:     by (intro nn_integral_cong) (simp add: field_simps)
hoelzl@57447:   also have "\<dots> = ereal (?F k a - ?F k 0)"
hoelzl@57447:   proof (rule nn_integral_FTC_Icc)
hoelzl@57235:     fix x assume "x \<in> {0..a}"
hoelzl@57235:     show "DERIV (?F k) x :> ?f k x"
hoelzl@57235:     proof(induction k)
hoelzl@57235:       case 0 show ?case by (auto intro!: derivative_eq_intros)
hoelzl@57235:     next
hoelzl@57235:       case (Suc k)
hoelzl@57235:       have "DERIV (\<lambda>x. ?F k x - (x^Suc k * exp (-x)) / fact (Suc k)) x :>
hoelzl@57235:         ?f k x - ((real (Suc k) - x) * x ^ k * exp (- x)) / real (fact (Suc k))"
hoelzl@57235:         by (intro DERIV_diff Suc)
hoelzl@57235:            (auto intro!: derivative_eq_intros simp del: fact_Suc power_Suc
hoelzl@57235:                  simp add: field_simps power_Suc[symmetric] real_of_nat_def[symmetric])
hoelzl@57235:       also have "(\<lambda>x. ?F k x - (x^Suc k * exp (-x)) / fact (Suc k)) = ?F (Suc k)"
hoelzl@57235:         by simp
hoelzl@57235:       also have "?f k x - ((real (Suc k) - x) * x ^ k * exp (- x)) / real (fact (Suc k)) = ?f (Suc k) x"
hoelzl@57235:         by (auto simp: field_simps simp del: fact_Suc)
hoelzl@57235:            (simp_all add: real_of_nat_Suc field_simps)
hoelzl@57235:       finally show ?case .
hoelzl@57235:     qed
hoelzl@57235:   qed auto
hoelzl@57235:   also have "\<dots> = ereal (1 - (\<Sum>n\<le>k. (a^n * exp (-a)) / fact n))"
haftmann@57418:     by (auto simp: power_0_left if_distrib[where f="\<lambda>x. x / a" for a] setsum.If_cases)
hoelzl@57235:   finally show ?thesis
hoelzl@57235:     by (cases "?I") (auto simp: field_simps)
hoelzl@57235: qed
hoelzl@57235: 
hoelzl@57235: lemma nn_intergal_power_times_exp_Ici:
hoelzl@57235:   shows "(\<integral>\<^sup>+x. ereal (x^k * exp (-x)) * indicator {0 ..} x \<partial>lborel) = fact k"
hoelzl@57235: proof (rule LIMSEQ_unique)
hoelzl@57235:   let ?X = "\<lambda>n. \<integral>\<^sup>+ x. ereal (x^k * exp (-x)) * indicator {0 .. real n} x \<partial>lborel"
hoelzl@57235:   show "?X ----> (\<integral>\<^sup>+x. ereal (x^k * exp (-x)) * indicator {0 ..} x \<partial>lborel)"
hoelzl@57235:     apply (intro nn_integral_LIMSEQ)
hoelzl@57235:     apply (auto simp: incseq_def le_fun_def eventually_sequentially
hoelzl@57235:                 split: split_indicator intro!: Lim_eventually)
hoelzl@57235:     apply (metis natceiling_le_eq)
hoelzl@57235:     done
hoelzl@57235: 
hoelzl@57235:   have "((\<lambda>x. (1 - (\<Sum>n\<le>k. (x ^ n / exp x) / real (fact n))) * fact k) ---> (1 - (\<Sum>n\<le>k. 0 / real (fact n))) * fact k) at_top"
hoelzl@57235:     by (intro tendsto_intros tendsto_power_div_exp_0) simp
hoelzl@57235:   then show "?X ----> fact k"
hoelzl@57235:     by (subst nn_intergal_power_times_exp_Icc)
hoelzl@57235:        (auto simp: exp_minus field_simps intro!: filterlim_compose[OF _ filterlim_real_sequentially])
hoelzl@57235: qed
hoelzl@57235: 
hoelzl@57235: definition erlang_density :: "nat \<Rightarrow> real \<Rightarrow> real \<Rightarrow> real" where
hoelzl@57235:   "erlang_density k l x = (if x < 0 then 0 else (l^(Suc k) * x^k * exp (- l * x)) / fact k)"
hoelzl@57235: 
hoelzl@57235: definition erlang_CDF ::  "nat \<Rightarrow> real \<Rightarrow> real \<Rightarrow> real" where
hoelzl@57235:   "erlang_CDF k l x = (if x < 0 then 0 else 1 - (\<Sum>n\<le>k. ((l * x)^n * exp (- l * x) / fact n)))"
hoelzl@57235: 
hoelzl@57235: lemma erlang_density_nonneg: "0 \<le> l \<Longrightarrow> 0 \<le> erlang_density k l x"
hoelzl@57235:   by (simp add: erlang_density_def)
hoelzl@57235: 
hoelzl@57235: lemma borel_measurable_erlang_density[measurable]: "erlang_density k l \<in> borel_measurable borel"
hoelzl@57235:   by (auto simp add: erlang_density_def[abs_def])
hoelzl@57235: 
hoelzl@57235: lemma erlang_CDF_transform: "0 < l \<Longrightarrow> erlang_CDF k l a = erlang_CDF k 1 (l * a)"
hoelzl@57235:   by (auto simp add: erlang_CDF_def mult_less_0_iff)
hoelzl@57235: 
hoelzl@57235: lemma nn_integral_erlang_density:
hoelzl@57235:   assumes [arith]: "0 < l"
hoelzl@57235:   shows "(\<integral>\<^sup>+ x. ereal (erlang_density k l x) * indicator {.. a} x \<partial>lborel) = erlang_CDF k l a"
hoelzl@57235: proof cases
hoelzl@57235:   assume [arith]: "0 \<le> a"
hoelzl@57235:   have eq: "\<And>x. indicator {0..a} (x / l) = indicator {0..a*l} x"
hoelzl@57235:     by (simp add: field_simps split: split_indicator)
hoelzl@57235:   have "(\<integral>\<^sup>+x. ereal (erlang_density k l x) * indicator {.. a} x \<partial>lborel) =
hoelzl@57235:     (\<integral>\<^sup>+x. (l/fact k) * (ereal ((l*x)^k * exp (- (l*x))) * indicator {0 .. a} x) \<partial>lborel)"
hoelzl@57235:     by (intro nn_integral_cong) (auto simp: erlang_density_def power_mult_distrib split: split_indicator)
hoelzl@57235:   also have "\<dots> = (l/fact k) * (\<integral>\<^sup>+x. ereal ((l*x)^k * exp (- (l*x))) * indicator {0 .. a} x \<partial>lborel)"
hoelzl@57235:     by (intro nn_integral_cmult) auto
hoelzl@57235:   also have "\<dots> = ereal (l/fact k) * ((1/l) * (\<integral>\<^sup>+x. ereal (x^k * exp (- x)) * indicator {0 .. l * a} x \<partial>lborel))"
hoelzl@57235:     by (subst nn_integral_real_affine[where c="1 / l" and t=0]) (auto simp: field_simps eq)
hoelzl@57235:   also have "\<dots> = (1 - (\<Sum>n\<le>k. ((l * a)^n * exp (-(l * a))) / fact n))"
hoelzl@57235:     by (subst nn_intergal_power_times_exp_Icc) auto
hoelzl@57235:   also have "\<dots> = erlang_CDF k l a"
hoelzl@57235:     by (auto simp: erlang_CDF_def)
hoelzl@57235:   finally show ?thesis .
hoelzl@57235: next
hoelzl@57235:   assume "\<not> 0 \<le> a" 
hoelzl@57235:   moreover then have "(\<integral>\<^sup>+ x. ereal (erlang_density k l x) * indicator {.. a} x \<partial>lborel) = (\<integral>\<^sup>+x. 0 \<partial>(lborel::real measure))"
hoelzl@57235:     by (intro nn_integral_cong) (auto simp: erlang_density_def)
hoelzl@57235:   ultimately show ?thesis
hoelzl@57235:     by (simp add: erlang_CDF_def)
hoelzl@57235: qed
hoelzl@57235: 
hoelzl@57235: lemma emeasure_erlang_density: 
hoelzl@57235:   "0 < l \<Longrightarrow> emeasure (density lborel (erlang_density k l)) {.. a} = erlang_CDF k l a"
hoelzl@57235:   by (simp add: emeasure_density nn_integral_erlang_density)
hoelzl@57235: 
hoelzl@57235: lemma nn_integral_erlang_ith_moment: 
hoelzl@57235:   fixes k i :: nat and l :: real
hoelzl@57235:   assumes [arith]: "0 < l" 
hoelzl@57235:   shows "(\<integral>\<^sup>+ x. ereal (erlang_density k l x * x ^ i) \<partial>lborel) = fact (k + i) / (fact k * l ^ i)"
hoelzl@57235: proof -
hoelzl@57235:   have eq: "\<And>x. indicator {0..} (x / l) = indicator {0..} x"
hoelzl@57235:     by (simp add: field_simps split: split_indicator)
hoelzl@57235:   have "(\<integral>\<^sup>+ x. ereal (erlang_density k l x * x^i) \<partial>lborel) =
hoelzl@57235:     (\<integral>\<^sup>+x. (l/(fact k * l^i)) * (ereal ((l*x)^(k+i) * exp (- (l*x))) * indicator {0 ..} x) \<partial>lborel)"
hoelzl@57235:     by (intro nn_integral_cong) (auto simp: erlang_density_def power_mult_distrib power_add split: split_indicator)
hoelzl@57235:   also have "\<dots> = (l/(fact k * l^i)) * (\<integral>\<^sup>+x. ereal ((l*x)^(k+i) * exp (- (l*x))) * indicator {0 ..} x \<partial>lborel)"
hoelzl@57235:     by (intro nn_integral_cmult) auto
hoelzl@57235:   also have "\<dots> = ereal (l/(fact k * l^i)) * ((1/l) * (\<integral>\<^sup>+x. ereal (x^(k+i) * exp (- x)) * indicator {0 ..} x \<partial>lborel))"
hoelzl@57235:     by (subst nn_integral_real_affine[where c="1 / l" and t=0]) (auto simp: field_simps eq)
hoelzl@57235:   also have "\<dots> = fact (k + i) / (fact k * l ^ i)"
hoelzl@57235:     by (subst nn_intergal_power_times_exp_Ici) auto
hoelzl@57235:   finally show ?thesis .
hoelzl@57235: qed
hoelzl@57235: 
hoelzl@57235: lemma prob_space_erlang_density:
hoelzl@57235:   assumes l[arith]: "0 < l"
hoelzl@57235:   shows "prob_space (density lborel (erlang_density k l))" (is "prob_space ?D")
hoelzl@57235: proof
hoelzl@57235:   show "emeasure ?D (space ?D) = 1"
hoelzl@57235:     using nn_integral_erlang_ith_moment[OF l, where k=k and i=0] by (simp add: emeasure_density)
hoelzl@57235: qed
hoelzl@57235: 
hoelzl@57235: lemma (in prob_space) erlang_distributed_le:
hoelzl@57235:   assumes D: "distributed M lborel X (erlang_density k l)"
hoelzl@57235:   assumes [simp, arith]: "0 < l" "0 \<le> a"
hoelzl@57235:   shows "\<P>(x in M. X x \<le> a) = erlang_CDF k l a"
hoelzl@57235: proof -
hoelzl@57235:   have "emeasure M {x \<in> space M. X x \<le> a } = emeasure (distr M lborel X) {.. a}"
hoelzl@57235:     using distributed_measurable[OF D]
hoelzl@57235:     by (subst emeasure_distr) (auto intro!: arg_cong2[where f=emeasure])
hoelzl@57235:   also have "\<dots> = emeasure (density lborel (erlang_density k l)) {.. a}"
hoelzl@57235:     unfolding distributed_distr_eq_density[OF D] ..
hoelzl@57235:   also have "\<dots> = erlang_CDF k l a"
hoelzl@57235:     by (auto intro!: emeasure_erlang_density)
hoelzl@57235:   finally show ?thesis
hoelzl@57235:     by (auto simp: measure_def)
hoelzl@57235: qed
hoelzl@57235: 
hoelzl@57235: lemma (in prob_space) erlang_distributed_gt:
hoelzl@57235:   assumes D[simp]: "distributed M lborel X (erlang_density k l)"
hoelzl@57235:   assumes [arith]: "0 < l" "0 \<le> a"
hoelzl@57235:   shows "\<P>(x in M. a < X x ) = 1 - (erlang_CDF k l a)"
hoelzl@57235: proof -
hoelzl@57235:   have " 1 - (erlang_CDF k l a) = 1 - \<P>(x in M. X x \<le> a)" by (subst erlang_distributed_le) auto
hoelzl@57235:   also have "\<dots> = prob (space M - {x \<in> space M. X x \<le> a })"
hoelzl@57235:     using distributed_measurable[OF D] by (auto simp: prob_compl)
hoelzl@57235:   also have "\<dots> = \<P>(x in M. a < X x )" by (auto intro!: arg_cong[where f=prob] simp: not_le)
hoelzl@57235:   finally show ?thesis by simp
hoelzl@57235: qed
hoelzl@57235: 
hoelzl@57235: lemma erlang_CDF_at0: "erlang_CDF k l 0 = 0"
hoelzl@57235:   by (induction k) (auto simp: erlang_CDF_def)
hoelzl@57235: 
hoelzl@57235: lemma erlang_distributedI:
hoelzl@57235:   assumes X[measurable]: "X \<in> borel_measurable M" and [arith]: "0 < l"
hoelzl@57235:     and X_distr: "\<And>a. 0 \<le> a \<Longrightarrow> emeasure M {x\<in>space M. X x \<le> a} = erlang_CDF k l a"
hoelzl@57235:   shows "distributed M lborel X (erlang_density k l)"
hoelzl@57235: proof (rule distributedI_borel_atMost)
hoelzl@57235:   fix a :: real
hoelzl@57235:   { assume "a \<le> 0"  
hoelzl@57235:     with X have "emeasure M {x\<in>space M. X x \<le> a} \<le> emeasure M {x\<in>space M. X x \<le> 0}"
hoelzl@57235:       by (intro emeasure_mono) auto
hoelzl@57235:     also have "... = 0"  by (auto intro!: erlang_CDF_at0 simp: X_distr[of 0])
hoelzl@57235:     finally have "emeasure M {x\<in>space M. X x \<le> a} \<le> 0" by simp
hoelzl@57235:     then have "emeasure M {x\<in>space M. X x \<le> a} = 0" by (simp add:emeasure_le_0_iff)
hoelzl@57235:   }
hoelzl@57235:   note eq_0 = this
hoelzl@57235: 
hoelzl@57235:   show "(\<integral>\<^sup>+ x. erlang_density k l x * indicator {..a} x \<partial>lborel) = ereal (erlang_CDF k l a)"
hoelzl@57235:     using nn_integral_erlang_density[of l k a]
hoelzl@57235:     by (simp add: times_ereal.simps(1)[symmetric] ereal_indicator del: times_ereal.simps)
hoelzl@57235: 
hoelzl@57235:   show "emeasure M {x\<in>space M. X x \<le> a} = ereal (erlang_CDF k l a)"
hoelzl@57235:     using X_distr[of a] eq_0 by (auto simp: one_ereal_def erlang_CDF_def)
hoelzl@57235: qed (simp_all add: erlang_density_nonneg)
hoelzl@57235: 
hoelzl@57235: lemma (in prob_space) erlang_distributed_iff:
hoelzl@57235:   assumes [arith]: "0<l"
hoelzl@57235:   shows "distributed M lborel X (erlang_density k l) \<longleftrightarrow>
hoelzl@57235:     (X \<in> borel_measurable M \<and> 0 < l \<and>  (\<forall>a\<ge>0. \<P>(x in M. X x \<le> a) = erlang_CDF k l a ))"
hoelzl@57235:   using
hoelzl@57235:     distributed_measurable[of M lborel X "erlang_density k l"]
hoelzl@57235:     emeasure_erlang_density[of l]
hoelzl@57235:     erlang_distributed_le[of X k l]
hoelzl@57235:   by (auto intro!: erlang_distributedI simp: one_ereal_def emeasure_eq_measure) 
hoelzl@57235: 
hoelzl@57235: lemma (in prob_space) erlang_distributed_mult_const:
hoelzl@57235:   assumes erlX: "distributed M lborel X (erlang_density k l)"
hoelzl@57235:   assumes a_pos[arith]: "0 < \<alpha>"  "0 < l"
hoelzl@57235:   shows  "distributed M lborel (\<lambda>x. \<alpha> * X x) (erlang_density k (l / \<alpha>))"
hoelzl@57235: proof (subst erlang_distributed_iff, safe)
hoelzl@57235:   have [measurable]: "random_variable borel X"  and  [arith]: "0 < l " 
hoelzl@57235:   and  [simp]: "\<And>a. 0 \<le> a \<Longrightarrow> prob {x \<in> space M. X x \<le> a} = erlang_CDF k l a"
hoelzl@57235:     by(insert erlX, auto simp: erlang_distributed_iff)
hoelzl@57235: 
hoelzl@57235:   show "random_variable borel (\<lambda>x. \<alpha> * X x)" "0 < l / \<alpha>"  "0 < l / \<alpha>" 
hoelzl@57235:     by (auto simp:field_simps)
hoelzl@57235:   
hoelzl@57235:   fix a:: real assume [arith]: "0 \<le> a"
hoelzl@57235:   obtain b:: real  where [simp, arith]: "b = a/ \<alpha>" by blast 
hoelzl@57235: 
hoelzl@57235:   have [arith]: "0 \<le> b" by (auto simp: divide_nonneg_pos)
hoelzl@57235:  
hoelzl@57235:   have "prob {x \<in> space M. \<alpha> * X x \<le> a}  = prob {x \<in> space M.  X x \<le> b}"
hoelzl@57235:     by (rule arg_cong[where f= prob]) (auto simp:field_simps)
hoelzl@57235:   
hoelzl@57235:   moreover have "prob {x \<in> space M. X x \<le> b} = erlang_CDF k l b" by auto
hoelzl@57235:   moreover have "erlang_CDF k (l / \<alpha>) a = erlang_CDF k l b" unfolding erlang_CDF_def by auto
hoelzl@57235:   ultimately show "prob {x \<in> space M. \<alpha> * X x \<le> a} = erlang_CDF k (l / \<alpha>) a" by fastforce  
hoelzl@57235: qed
hoelzl@57235: 
hoelzl@57235: lemma (in prob_space) has_bochner_integral_erlang_ith_moment:
hoelzl@57235:   fixes k i :: nat and l :: real
hoelzl@57235:   assumes [arith]: "0 < l" and D: "distributed M lborel X (erlang_density k l)"
hoelzl@57235:   shows "has_bochner_integral M (\<lambda>x. X x ^ i) (fact (k + i) / (fact k * l ^ i))"
hoelzl@57235: proof (rule has_bochner_integral_nn_integral)
hoelzl@57235:   show "AE x in M. 0 \<le> X x ^ i"
hoelzl@57235:     by (subst distributed_AE2[OF D]) (auto simp: erlang_density_def)
hoelzl@57235:   show "(\<integral>\<^sup>+ x. ereal (X x ^ i) \<partial>M) = ereal (fact (k + i) / (fact k * l ^ i))"
hoelzl@57235:     using nn_integral_erlang_ith_moment[of l k i]
hoelzl@57235:     by (subst distributed_nn_integral[symmetric, OF D]) auto
hoelzl@57235: qed (insert distributed_measurable[OF D], simp)
hoelzl@57235: 
hoelzl@57235: lemma (in prob_space) erlang_ith_moment_integrable:
hoelzl@57235:   "0 < l \<Longrightarrow> distributed M lborel X (erlang_density k l) \<Longrightarrow> integrable M (\<lambda>x. X x ^ i)"
hoelzl@57235:   by rule (rule has_bochner_integral_erlang_ith_moment)
hoelzl@57235: 
hoelzl@57235: lemma (in prob_space) erlang_ith_moment:
hoelzl@57235:   "0 < l \<Longrightarrow> distributed M lborel X (erlang_density k l) \<Longrightarrow>
hoelzl@57235:     expectation (\<lambda>x. X x ^ i) = fact (k + i) / (fact k * l ^ i)"
hoelzl@57235:   by (rule has_bochner_integral_integral_eq) (rule has_bochner_integral_erlang_ith_moment)
hoelzl@57235: 
hoelzl@57235: lemma (in prob_space) erlang_distributed_variance:
hoelzl@57235:   assumes [arith]: "0 < l" and "distributed M lborel X (erlang_density k l)"
hoelzl@57235:   shows "variance X = (k + 1) / l\<^sup>2"
hoelzl@57235: proof (subst variance_eq)
hoelzl@57235:   show "integrable M X" "integrable M (\<lambda>x. (X x)\<^sup>2)"
hoelzl@57235:     using erlang_ith_moment_integrable[OF assms, of 1] erlang_ith_moment_integrable[OF assms, of 2]
hoelzl@57235:     by auto
hoelzl@57235: 
hoelzl@57235:   show "expectation (\<lambda>x. (X x)\<^sup>2) - (expectation X)\<^sup>2 = real (k + 1) / l\<^sup>2"
hoelzl@57235:     using erlang_ith_moment[OF assms, of 1] erlang_ith_moment[OF assms, of 2]
hoelzl@57235:     by simp (auto simp: power2_eq_square field_simps real_of_nat_Suc)
hoelzl@57235: qed
hoelzl@57235: 
hoelzl@50419: subsection {* Exponential distribution *}
hoelzl@50419: 
hoelzl@57235: abbreviation exponential_density :: "real \<Rightarrow> real \<Rightarrow> real" where
hoelzl@57235:   "exponential_density \<equiv> erlang_density 0"
hoelzl@50419: 
hoelzl@57235: lemma exponential_density_def:
hoelzl@57235:   "exponential_density l x = (if x < 0 then 0 else l * exp (- x * l))"
hoelzl@57235:   by (simp add: fun_eq_iff erlang_density_def)
hoelzl@57235: 
hoelzl@57235: lemma erlang_CDF_0: "erlang_CDF 0 l a = (if 0 \<le> a then 1 - exp (- l * a) else 0)"
hoelzl@57235:   by (simp add: erlang_CDF_def)
hoelzl@50419: 
hoelzl@50419: lemma (in prob_space) exponential_distributed_params:
hoelzl@50419:   assumes D: "distributed M lborel X (exponential_density l)"
hoelzl@50419:   shows "0 < l"
hoelzl@50419: proof (cases l "0 :: real" rule: linorder_cases)
hoelzl@50419:   assume "l < 0"
hoelzl@50419:   have "emeasure lborel {0 <.. 1::real} \<le>
hoelzl@50419:     emeasure lborel {x :: real \<in> space lborel. 0 < x}"
hoelzl@50419:     by (rule emeasure_mono) (auto simp: greaterThan_def[symmetric])
hoelzl@50419:   also have "emeasure lborel {x :: real \<in> space lborel. 0 < x} = 0"
hoelzl@50419:   proof -
hoelzl@50419:     have "AE x in lborel. 0 \<le> exponential_density l x"
hoelzl@50419:       using assms by (auto simp: distributed_real_AE)
hoelzl@50419:     then have "AE x in lborel. x \<le> (0::real)"
hoelzl@50419:       apply eventually_elim 
hoelzl@50419:       using `l < 0`
hoelzl@50419:       apply (auto simp: exponential_density_def zero_le_mult_iff split: split_if_asm)
hoelzl@50419:       done
hoelzl@50419:     then show "emeasure lborel {x :: real \<in> space lborel. 0 < x} = 0"
hoelzl@50419:       by (subst (asm) AE_iff_measurable[OF _ refl]) (auto simp: not_le greaterThan_def[symmetric])
hoelzl@50419:   qed
hoelzl@50419:   finally show "0 < l" by simp
hoelzl@50419: next
hoelzl@50419:   assume "l = 0"
hoelzl@50419:   then have [simp]: "\<And>x. ereal (exponential_density l x) = 0"
hoelzl@50419:     by (simp add: exponential_density_def)
hoelzl@50419:   interpret X: prob_space "distr M lborel X"
hoelzl@50419:     using distributed_measurable[OF D] by (rule prob_space_distr)
hoelzl@50419:   from X.emeasure_space_1
hoelzl@50419:   show "0 < l"
hoelzl@50419:     by (simp add: emeasure_density distributed_distr_eq_density[OF D])
hoelzl@50419: qed assumption
hoelzl@50419: 
hoelzl@57235: lemma prob_space_exponential_density: "0 < l \<Longrightarrow> prob_space (density lborel (exponential_density l))"
hoelzl@57235:   by (rule prob_space_erlang_density)
hoelzl@50419: 
hoelzl@50419: lemma (in prob_space) exponential_distributedD_le:
hoelzl@50419:   assumes D: "distributed M lborel X (exponential_density l)" and a: "0 \<le> a"
hoelzl@50419:   shows "\<P>(x in M. X x \<le> a) = 1 - exp (- a * l)"
hoelzl@57235:   using erlang_distributed_le[OF D exponential_distributed_params[OF D] a] a
hoelzl@57235:   by (simp add: erlang_CDF_def)
hoelzl@50419: 
hoelzl@50419: lemma (in prob_space) exponential_distributedD_gt:
hoelzl@50419:   assumes D: "distributed M lborel X (exponential_density l)" and a: "0 \<le> a"
hoelzl@50419:   shows "\<P>(x in M. a < X x ) = exp (- a * l)"
hoelzl@57235:   using erlang_distributed_gt[OF D exponential_distributed_params[OF D] a] a
hoelzl@57235:   by (simp add: erlang_CDF_def)
hoelzl@50419: 
hoelzl@50419: lemma (in prob_space) exponential_distributed_memoryless:
hoelzl@50419:   assumes D: "distributed M lborel X (exponential_density l)" and a: "0 \<le> a"and t: "0 \<le> t"
hoelzl@50419:   shows "\<P>(x in M. a + t < X x \<bar> a < X x) = \<P>(x in M. t < X x)"
hoelzl@50419: proof -
hoelzl@50419:   have "\<P>(x in M. a + t < X x \<bar> a < X x) = \<P>(x in M. a + t < X x) / \<P>(x in M. a < X x)"
hoelzl@50419:     using `0 \<le> t` by (auto simp: cond_prob_def intro!: arg_cong[where f=prob] arg_cong2[where f="op /"])
hoelzl@50419:   also have "\<dots> = exp (- (a + t) * l) / exp (- a * l)"
hoelzl@50419:     using a t by (simp add: exponential_distributedD_gt[OF D])
hoelzl@50419:   also have "\<dots> = exp (- t * l)"
hoelzl@50419:     using exponential_distributed_params[OF D] by (auto simp: field_simps exp_add[symmetric])
hoelzl@50419:   finally show ?thesis
hoelzl@50419:     using t by (simp add: exponential_distributedD_gt[OF D])
hoelzl@50419: qed
hoelzl@50419: 
hoelzl@50419: lemma exponential_distributedI:
hoelzl@50419:   assumes X[measurable]: "X \<in> borel_measurable M" and [arith]: "0 < l"
hoelzl@50419:     and X_distr: "\<And>a. 0 \<le> a \<Longrightarrow> emeasure M {x\<in>space M. X x \<le> a} = 1 - exp (- a * l)"
hoelzl@50419:   shows "distributed M lborel X (exponential_density l)"
hoelzl@57235: proof (rule erlang_distributedI)
hoelzl@57235:   fix a :: real assume "0 \<le> a" then show "emeasure M {x \<in> space M. X x \<le> a} = ereal (erlang_CDF 0 l a)"
hoelzl@57235:     using X_distr[of a] by (simp add: erlang_CDF_def one_ereal_def)
hoelzl@57235: qed fact+
hoelzl@50419: 
hoelzl@50419: lemma (in prob_space) exponential_distributed_iff:
hoelzl@50419:   "distributed M lborel X (exponential_density l) \<longleftrightarrow>
hoelzl@50419:     (X \<in> borel_measurable M \<and> 0 < l \<and> (\<forall>a\<ge>0. \<P>(x in M. X x \<le> a) = 1 - exp (- a * l)))"
hoelzl@57235:   using exponential_distributed_params[of X l] erlang_distributed_iff[of l X 0] by (auto simp: erlang_CDF_0)
hoelzl@50419: 
hoelzl@50419: 
hoelzl@50419: lemma (in prob_space) exponential_distributed_expectation:
hoelzl@57235:   "distributed M lborel X (exponential_density l) \<Longrightarrow> expectation X = 1 / l"
hoelzl@57235:   using erlang_ith_moment[OF exponential_distributed_params, of X l X 0 1] by simp
hoelzl@57235: 
hoelzl@57235: lemma exponential_density_nonneg: "0 < l \<Longrightarrow> 0 \<le> exponential_density l x"
hoelzl@57235:   by (auto simp: exponential_density_def)
hoelzl@57235: 
hoelzl@57235: lemma (in prob_space) exponential_distributed_min:
hoelzl@57235:   assumes expX: "distributed M lborel X (exponential_density l)"
hoelzl@57235:   assumes expY: "distributed M lborel Y (exponential_density u)"
hoelzl@57235:   assumes ind: "indep_var borel X borel Y"
hoelzl@57235:   shows "distributed M lborel (\<lambda>x. min (X x) (Y x)) (exponential_density (l + u))"
hoelzl@57235: proof (subst exponential_distributed_iff, safe)
hoelzl@57235:   have randX: "random_variable borel X" using expX by (simp add: exponential_distributed_iff)
hoelzl@57235:   moreover have randY: "random_variable borel Y" using expY by (simp add: exponential_distributed_iff)
hoelzl@57235:   ultimately show "random_variable borel (\<lambda>x. min (X x) (Y x))" by auto
hoelzl@57235:   
hoelzl@57235:   have "0 < l" by (rule exponential_distributed_params) fact
hoelzl@57235:   moreover have "0 < u" by (rule exponential_distributed_params) fact
hoelzl@57235:   ultimately  show " 0 < l + u" by force
hoelzl@57235: 
hoelzl@57235:   fix a::real assume a[arith]: "0 \<le> a"
hoelzl@57235:   have gt1[simp]: "\<P>(x in M. a < X x ) = exp (- a * l)" by (rule exponential_distributedD_gt[OF expX a]) 
hoelzl@57235:   have gt2[simp]: "\<P>(x in M. a < Y x ) = exp (- a * u)" by (rule exponential_distributedD_gt[OF expY a]) 
hoelzl@57235: 
hoelzl@57235:   have "\<P>(x in M. a < (min (X x) (Y x)) ) =  \<P>(x in M. a < (X x) \<and> a < (Y x))" by (auto intro!:arg_cong[where f=prob])
hoelzl@57235: 
hoelzl@57235:   also have " ... =  \<P>(x in M. a < (X x)) *  \<P>(x in M. a< (Y x) )"
hoelzl@57235:     using prob_indep_random_variable[OF ind, of "{a <..}" "{a <..}"] by simp
hoelzl@57235:   also have " ... = exp (- a * (l + u))" by (auto simp:field_simps mult_exp_exp)
hoelzl@57235:   finally have indep_prob: "\<P>(x in M. a < (min (X x) (Y x)) ) = exp (- a * (l + u))" .
hoelzl@57235: 
hoelzl@57235:   have "{x \<in> space M. (min (X x) (Y x)) \<le>a } = (space M - {x \<in> space M. a<(min (X x) (Y x)) })"
hoelzl@57235:     by auto
hoelzl@57235:   then have "1 - prob {x \<in> space M. a < (min (X x) (Y x))} = prob {x \<in> space M. (min (X x) (Y x)) \<le> a}"
hoelzl@57235:     using randX randY by (auto simp: prob_compl) 
hoelzl@57235:   then show "prob {x \<in> space M. (min (X x) (Y x)) \<le> a} = 1 - exp (- a * (l + u))"
hoelzl@57235:     using indep_prob by auto
hoelzl@57235: qed
hoelzl@57235:  
hoelzl@57235: lemma (in prob_space) exponential_distributed_Min:
hoelzl@57235:   assumes finI: "finite I"
hoelzl@57235:   assumes A: "I \<noteq> {}"
hoelzl@57235:   assumes expX: "\<And>i. i \<in> I \<Longrightarrow> distributed M lborel (X i) (exponential_density (l i))"
hoelzl@57235:   assumes ind: "indep_vars (\<lambda>i. borel) X I" 
hoelzl@57235:   shows "distributed M lborel (\<lambda>x. Min ((\<lambda>i. X i x)`I)) (exponential_density (\<Sum>i\<in>I. l i))"
hoelzl@57235: using assms
hoelzl@57235: proof (induct rule: finite_ne_induct)
hoelzl@57235:   case (singleton i) then show ?case by simp
hoelzl@57235: next
hoelzl@57235:   case (insert i I)
hoelzl@57235:   then have "distributed M lborel (\<lambda>x. min (X i x) (Min ((\<lambda>i. X i x)`I))) (exponential_density (l i + (\<Sum>i\<in>I. l i)))"
hoelzl@57235:       by (intro exponential_distributed_min indep_vars_Min insert)
hoelzl@57235:          (auto intro: indep_vars_subset) 
hoelzl@57235:   then show ?case
hoelzl@57235:     using insert by simp
hoelzl@57235: qed
hoelzl@57235: 
hoelzl@57235: lemma (in prob_space) exponential_distributed_variance:
hoelzl@57235:   "distributed M lborel X (exponential_density l) \<Longrightarrow> variance X = 1 / l\<^sup>2"
hoelzl@57235:   using erlang_distributed_variance[OF exponential_distributed_params, of X l X 0] by simp
hoelzl@57235: 
hoelzl@57235: lemma nn_integral_zero': "AE x in M. f x = 0 \<Longrightarrow> (\<integral>\<^sup>+x. f x \<partial>M) = 0"
hoelzl@57235:   by (simp cong: nn_integral_cong_AE)
hoelzl@57235: 
hoelzl@57235: lemma convolution_erlang_density:
hoelzl@57235:   fixes k\<^sub>1 k\<^sub>2 :: nat
hoelzl@57235:   assumes [simp, arith]: "0 < l"
hoelzl@57235:   shows "(\<lambda>x. \<integral>\<^sup>+y. ereal (erlang_density k\<^sub>1 l (x - y)) * ereal (erlang_density k\<^sub>2 l y) \<partial>lborel) =
hoelzl@57235:     (erlang_density (Suc k\<^sub>1 + Suc k\<^sub>2 - 1) l)"
hoelzl@57235:       (is "?LHS = ?RHS")
hoelzl@57235: proof
hoelzl@57235:   fix x :: real
hoelzl@57235:   have "x \<le> 0 \<or> 0 < x"
hoelzl@57235:     by arith
hoelzl@57235:   then show "?LHS x = ?RHS x"
hoelzl@57235:   proof
hoelzl@57235:     assume "x \<le> 0" then show ?thesis
hoelzl@57235:       apply (subst nn_integral_zero')
hoelzl@57235:       apply (rule AE_I[where N="{0}"])
hoelzl@57235:       apply (auto simp add: erlang_density_def not_less)
hoelzl@57235:       done
hoelzl@57235:   next
hoelzl@57235:     note zero_le_mult_iff[simp] zero_le_divide_iff[simp]
hoelzl@57235:   
hoelzl@57235:     have I_eq1: "integral\<^sup>N lborel (erlang_density (Suc k\<^sub>1 + Suc k\<^sub>2 - 1) l) = 1"
hoelzl@57235:       using nn_integral_erlang_ith_moment[of l "Suc k\<^sub>1 + Suc k\<^sub>2 - 1" 0] by (simp del: fact_Suc)
hoelzl@57235:   
hoelzl@57235:     have 1: "(\<integral>\<^sup>+ x. ereal (erlang_density (Suc k\<^sub>1 + Suc k\<^sub>2 - 1) l x * indicator {0<..} x) \<partial>lborel) = 1"
hoelzl@57235:       apply (subst I_eq1[symmetric])
hoelzl@57235:       unfolding erlang_density_def
hoelzl@57235:       by (auto intro!: nn_integral_cong split:split_indicator)
hoelzl@57235:   
hoelzl@57235:     have "prob_space (density lborel ?LHS)"
hoelzl@57235:       unfolding times_ereal.simps[symmetric]
hoelzl@57235:       by (intro prob_space_convolution_density) 
hoelzl@57235:          (auto intro!: prob_space_erlang_density erlang_density_nonneg)
hoelzl@57235:     then have 2: "integral\<^sup>N lborel ?LHS = 1"
hoelzl@57235:       by (auto dest!: prob_space.emeasure_space_1 simp: emeasure_density)
hoelzl@57235:   
hoelzl@57235:     let ?I = "(integral\<^sup>N lborel (\<lambda>y. ereal ((1 - y)^ k\<^sub>1 * y^k\<^sub>2 * indicator {0..1} y)))"
hoelzl@57235:     let ?C = "real (fact (Suc (k\<^sub>1 + k\<^sub>2))) / (real (fact k\<^sub>1) * real (fact k\<^sub>2))"
hoelzl@57235:     let ?s = "Suc k\<^sub>1 + Suc k\<^sub>2 - 1"
hoelzl@57235:     let ?L = "(\<lambda>x. \<integral>\<^sup>+y. ereal (erlang_density k\<^sub>1 l (x- y) * erlang_density k\<^sub>2 l y * indicator {0..x} y) \<partial>lborel)"
hoelzl@57235: 
hoelzl@57235:     { fix x :: real assume [arith]: "0 < x"
hoelzl@57235:       have *: "\<And>x y n. (x - y * x::real)^n = x^n * (1 - y)^n"
hoelzl@57235:         unfolding power_mult_distrib[symmetric] by (simp add: field_simps)
hoelzl@57235:     
hoelzl@57235:       have "?LHS x = ?L x"
hoelzl@57235:         unfolding erlang_density_def
hoelzl@57235:         by (auto intro!: nn_integral_cong split:split_indicator)
hoelzl@57235:       also have "... = (\<lambda>x. ereal ?C * ?I * erlang_density ?s l x) x"
hoelzl@57235:         apply (subst nn_integral_real_affine[where c=x and t = 0])
hoelzl@57235:         apply (simp_all add: nn_integral_cmult[symmetric] nn_integral_multc[symmetric] erlang_density_nonneg del: fact_Suc)
hoelzl@57235:         apply (intro nn_integral_cong)
hoelzl@57235:         apply (auto simp add: erlang_density_def mult_less_0_iff exp_minus field_simps exp_diff power_add *
hoelzl@57235:                     simp del: fact_Suc split: split_indicator)
hoelzl@57235:         done
hoelzl@57235:       finally have "(\<integral>\<^sup>+y. ereal (erlang_density k\<^sub>1 l (x - y) * erlang_density k\<^sub>2 l y) \<partial>lborel) = 
hoelzl@57235:         (\<lambda>x. ereal ?C * ?I * erlang_density ?s l x) x"
hoelzl@57235:         by simp }
hoelzl@57235:     note * = this
hoelzl@57235: 
hoelzl@57235:     assume [arith]: "0 < x"
hoelzl@57235:     have 3: "1 = integral\<^sup>N lborel (\<lambda>xa. ?LHS xa * indicator {0<..} xa)"
hoelzl@57235:       by (subst 2[symmetric])
hoelzl@57235:          (auto intro!: nn_integral_cong_AE AE_I[where N="{0}"]
hoelzl@57235:                simp: erlang_density_def  nn_integral_multc[symmetric] indicator_def split: split_if_asm)
hoelzl@57235:     also have "... = integral\<^sup>N lborel (\<lambda>x. (ereal (?C) * ?I) * ((erlang_density ?s l x) * indicator {0<..} x))"
hoelzl@57235:       by (auto intro!: nn_integral_cong simp: * split: split_indicator)
hoelzl@57235:     also have "... = ereal (?C) * ?I"
hoelzl@57235:       using 1
hoelzl@57235:       by (auto simp: nn_integral_nonneg nn_integral_cmult)  
hoelzl@57235:     finally have " ereal (?C) * ?I = 1" by presburger
hoelzl@57235:   
hoelzl@57235:     then show ?thesis
hoelzl@57235:       using * by simp
hoelzl@57235:   qed
hoelzl@57235: qed
hoelzl@57235: 
hoelzl@57235: lemma (in prob_space) sum_indep_erlang:
hoelzl@57235:   assumes indep: "indep_var borel X borel Y"
hoelzl@57235:   assumes [simp, arith]: "0 < l"
hoelzl@57235:   assumes erlX: "distributed M lborel X (erlang_density k\<^sub>1 l)"
hoelzl@57235:   assumes erlY: "distributed M lborel Y (erlang_density k\<^sub>2 l)"
hoelzl@57235:   shows "distributed M lborel (\<lambda>x. X x + Y x) (erlang_density (Suc k\<^sub>1 + Suc k\<^sub>2 - 1) l)"
hoelzl@57235:   using assms
hoelzl@57235:   apply (subst convolution_erlang_density[symmetric, OF `0<l`])
hoelzl@57235:   apply (intro distributed_convolution)
hoelzl@57235:   apply auto
hoelzl@57235:   done
hoelzl@57235: 
hoelzl@57235: lemma (in prob_space) erlang_distributed_setsum:
hoelzl@57235:   assumes finI : "finite I"
hoelzl@57235:   assumes A: "I \<noteq> {}"
hoelzl@57235:   assumes [simp, arith]: "0 < l"
hoelzl@57235:   assumes expX: "\<And>i. i \<in> I \<Longrightarrow> distributed M lborel (X i) (erlang_density (k i) l)"
hoelzl@57235:   assumes ind: "indep_vars (\<lambda>i. borel) X I"
hoelzl@57235:   shows "distributed M lborel (\<lambda>x. \<Sum>i\<in>I. X i x) (erlang_density ((\<Sum>i\<in>I. Suc (k i)) - 1) l)"
hoelzl@57235: using assms
hoelzl@57235: proof (induct rule: finite_ne_induct)
hoelzl@57235:   case (singleton i) then show ?case by auto
hoelzl@57235: next
hoelzl@57235:   case (insert i I)
hoelzl@57235:     then have "distributed M lborel (\<lambda>x. (X i x) + (\<Sum>i\<in> I. X i x)) (erlang_density (Suc (k i) + Suc ((\<Sum>i\<in>I. Suc (k i)) - 1) - 1) l)"
hoelzl@57235:       by(intro sum_indep_erlang indep_vars_setsum) (auto intro!: indep_vars_subset)
hoelzl@57235:     also have "(\<lambda>x. (X i x) + (\<Sum>i\<in> I. X i x)) = (\<lambda>x. \<Sum>i\<in>insert i I. X i x)"
hoelzl@57235:       using insert by auto
hoelzl@57235:     also have "Suc(k i) + Suc ((\<Sum>i\<in>I. Suc (k i)) - 1) - 1 = (\<Sum>i\<in>insert i I. Suc (k i)) - 1"
hoelzl@57235:       using insert by (auto intro!: Suc_pred simp: ac_simps)    
hoelzl@57235:     finally show ?case by fast
hoelzl@57235: qed
hoelzl@57235: 
hoelzl@57235: lemma (in prob_space) exponential_distributed_setsum:
hoelzl@57235:   assumes finI: "finite I"
hoelzl@57235:   assumes A: "I \<noteq> {}"
hoelzl@57235:   assumes expX: "\<And>i. i \<in> I \<Longrightarrow> distributed M lborel (X i) (exponential_density l)"
hoelzl@57235:   assumes ind: "indep_vars (\<lambda>i. borel) X I" 
hoelzl@57235:   shows "distributed M lborel (\<lambda>x. \<Sum>i\<in>I. X i x) (erlang_density ((card I) - 1) l)"
hoelzl@57235: proof -
hoelzl@57235:   obtain i where "i \<in> I" using assms by auto
hoelzl@57235:   note exponential_distributed_params[OF expX[OF this]]
hoelzl@57235:   from erlang_distributed_setsum[OF assms(1,2) this assms(3,4)] show ?thesis by simp
hoelzl@57235: qed
hoelzl@50419: 
hoelzl@57252: lemma (in information_space) entropy_exponential:
hoelzl@57252:   assumes D: "distributed M lborel X (exponential_density l)"
hoelzl@57252:   shows "entropy b lborel X = log b (exp 1 / l)"
hoelzl@57252: proof -
hoelzl@57252:   have l[simp, arith]: "0 < l" by (rule exponential_distributed_params[OF D])
hoelzl@57252:  
hoelzl@57252:   have [simp]: "integrable lborel (exponential_density l)"
hoelzl@57252:     using distributed_integrable[OF D, of "\<lambda>_. 1"] by simp
hoelzl@57252: 
hoelzl@57252:   have [simp]: "integral\<^sup>L lborel (exponential_density l) = 1"
hoelzl@57252:     using distributed_integral[OF D, of "\<lambda>_. 1"] by (simp add: prob_space)
hoelzl@57252:     
hoelzl@57252:   have [simp]: "integrable lborel (\<lambda>x. exponential_density l x * x)"
hoelzl@57252:     using erlang_ith_moment_integrable[OF l D, of 1] distributed_integrable[OF D, of "\<lambda>x. x"] by simp
hoelzl@57252: 
hoelzl@57252:   have [simp]: "integral\<^sup>L lborel (\<lambda>x. exponential_density l x * x) = 1 / l"
hoelzl@57252:     using erlang_ith_moment[OF l D, of 1] distributed_integral[OF D, of "\<lambda>x. x"] by simp
hoelzl@57252:     
hoelzl@57252:   have "entropy b lborel X = - (\<integral> x. exponential_density l x * log b (exponential_density l x) \<partial>lborel)"
hoelzl@57252:     using D by (rule entropy_distr)
hoelzl@57252:   also have "(\<integral> x. exponential_density l x * log b (exponential_density l x) \<partial>lborel) = 
hoelzl@57252:     (\<integral> x. (ln l * exponential_density l x - l * (exponential_density l x * x)) / ln b \<partial>lborel)"
hoelzl@57252:     by (intro integral_cong) (auto simp: log_def ln_mult exponential_density_def field_simps)
hoelzl@57252:   also have "\<dots> = (ln l - 1) / ln b"
hoelzl@57252:     by simp
hoelzl@57252:   finally show ?thesis
hoelzl@57252:     by (simp add: log_def divide_simps ln_div)
hoelzl@57252: qed
hoelzl@57252: 
hoelzl@50419: subsection {* Uniform distribution *}
hoelzl@50419: 
hoelzl@50419: lemma uniform_distrI:
hoelzl@50419:   assumes X: "X \<in> measurable M M'"
hoelzl@50419:     and A: "A \<in> sets M'" "emeasure M' A \<noteq> \<infinity>" "emeasure M' A \<noteq> 0"
hoelzl@50419:   assumes distr: "\<And>B. B \<in> sets M' \<Longrightarrow> emeasure M (X -` B \<inter> space M) = emeasure M' (A \<inter> B) / emeasure M' A"
hoelzl@50419:   shows "distr M M' X = uniform_measure M' A"
hoelzl@50419:   unfolding uniform_measure_def
hoelzl@50419: proof (intro measure_eqI)
hoelzl@50419:   let ?f = "\<lambda>x. indicator A x / emeasure M' A"
hoelzl@50419:   fix B assume B: "B \<in> sets (distr M M' X)"
hoelzl@50419:   with X have "emeasure M (X -` B \<inter> space M) = emeasure M' (A \<inter> B) / emeasure M' A"
hoelzl@50419:     by (simp add: distr[of B] measurable_sets)
hoelzl@50419:   also have "\<dots> = (1 / emeasure M' A) * emeasure M' (A \<inter> B)"
hoelzl@50419:      by simp
wenzelm@53015:   also have "\<dots> = (\<integral>\<^sup>+ x. (1 / emeasure M' A) * indicator (A \<inter> B) x \<partial>M')"
hoelzl@50419:     using A B
hoelzl@56996:     by (intro nn_integral_cmult_indicator[symmetric]) (auto intro!: zero_le_divide_ereal)
wenzelm@53015:   also have "\<dots> = (\<integral>\<^sup>+ x. ?f x * indicator B x \<partial>M')"
hoelzl@56996:     by (rule nn_integral_cong) (auto split: split_indicator)
hoelzl@50419:   finally show "emeasure (distr M M' X) B = emeasure (density M' ?f) B"
hoelzl@50419:     using A B X by (auto simp add: emeasure_distr emeasure_density)
hoelzl@50419: qed simp
hoelzl@50419: 
hoelzl@50419: lemma uniform_distrI_borel:
hoelzl@50419:   fixes A :: "real set"
hoelzl@50419:   assumes X[measurable]: "X \<in> borel_measurable M" and A: "emeasure lborel A = ereal r" "0 < r"
hoelzl@50419:     and [measurable]: "A \<in> sets borel"
hoelzl@50419:   assumes distr: "\<And>a. emeasure M {x\<in>space M. X x \<le> a} = emeasure lborel (A \<inter> {.. a}) / r"
hoelzl@50419:   shows "distributed M lborel X (\<lambda>x. indicator A x / measure lborel A)"
hoelzl@50419: proof (rule distributedI_borel_atMost)
hoelzl@50419:   let ?f = "\<lambda>x. 1 / r * indicator A x"
hoelzl@50419:   fix a
hoelzl@50419:   have "emeasure lborel (A \<inter> {..a}) \<le> emeasure lborel A"
hoelzl@50419:     using A by (intro emeasure_mono) auto
hoelzl@50419:   also have "\<dots> < \<infinity>"
hoelzl@50419:     using A by simp
hoelzl@50419:   finally have fin: "emeasure lborel (A \<inter> {..a}) \<noteq> \<infinity>"
hoelzl@50419:     by simp
hoelzl@50419:   from emeasure_eq_ereal_measure[OF this]
hoelzl@50419:   have fin_eq: "emeasure lborel (A \<inter> {..a}) / r = ereal (measure lborel (A \<inter> {..a}) / r)"
hoelzl@50419:     using A by simp
hoelzl@50419:   then show "emeasure M {x\<in>space M. X x \<le> a} = ereal (measure lborel (A \<inter> {..a}) / r)"
hoelzl@50419:     using distr by simp
hoelzl@50419:  
wenzelm@53015:   have "(\<integral>\<^sup>+ x. ereal (indicator A x / measure lborel A * indicator {..a} x) \<partial>lborel) =
wenzelm@53015:     (\<integral>\<^sup>+ x. ereal (1 / measure lborel A) * indicator (A \<inter> {..a}) x \<partial>lborel)"
hoelzl@56996:     by (auto intro!: nn_integral_cong split: split_indicator)
hoelzl@50419:   also have "\<dots> = ereal (1 / measure lborel A) * emeasure lborel (A \<inter> {..a})"
hoelzl@50419:     using `A \<in> sets borel`
hoelzl@56996:     by (intro nn_integral_cmult_indicator) (auto simp: measure_nonneg)
hoelzl@50419:   also have "\<dots> = ereal (measure lborel (A \<inter> {..a}) / r)"
hoelzl@50419:     unfolding emeasure_eq_ereal_measure[OF fin] using A by (simp add: measure_def)
wenzelm@53015:   finally show "(\<integral>\<^sup>+ x. ereal (indicator A x / measure lborel A * indicator {..a} x) \<partial>lborel) =
hoelzl@50419:     ereal (measure lborel (A \<inter> {..a}) / r)" .
hoelzl@56571: qed (auto simp: measure_nonneg)
hoelzl@50419: 
hoelzl@50419: lemma (in prob_space) uniform_distrI_borel_atLeastAtMost:
hoelzl@50419:   fixes a b :: real
hoelzl@50419:   assumes X: "X \<in> borel_measurable M" and "a < b"
hoelzl@50419:   assumes distr: "\<And>t. a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow> \<P>(x in M. X x \<le> t) = (t - a) / (b - a)"
hoelzl@50419:   shows "distributed M lborel X (\<lambda>x. indicator {a..b} x / measure lborel {a..b})"
hoelzl@50419: proof (rule uniform_distrI_borel)
hoelzl@50419:   fix t
hoelzl@50419:   have "t < a \<or> (a \<le> t \<and> t \<le> b) \<or> b < t"
hoelzl@50419:     by auto
hoelzl@50419:   then show "emeasure M {x\<in>space M. X x \<le> t} = emeasure lborel ({a .. b} \<inter> {..t}) / (b - a)"
hoelzl@50419:   proof (elim disjE conjE)
hoelzl@50419:     assume "t < a" 
hoelzl@50419:     then have "emeasure M {x\<in>space M. X x \<le> t} \<le> emeasure M {x\<in>space M. X x \<le> a}"
hoelzl@50419:       using X by (auto intro!: emeasure_mono measurable_sets)
hoelzl@50419:     also have "\<dots> = 0"
hoelzl@50419:       using distr[of a] `a < b` by (simp add: emeasure_eq_measure)
hoelzl@50419:     finally have "emeasure M {x\<in>space M. X x \<le> t} = 0"
hoelzl@50419:       by (simp add: antisym measure_nonneg emeasure_le_0_iff)
hoelzl@50419:     with `t < a` show ?thesis by simp
hoelzl@50419:   next
hoelzl@50419:     assume bnds: "a \<le> t" "t \<le> b"
hoelzl@50419:     have "{a..b} \<inter> {..t} = {a..t}"
hoelzl@50419:       using bnds by auto
hoelzl@50419:     then show ?thesis using `a \<le> t` `a < b`
hoelzl@50419:       using distr[OF bnds] by (simp add: emeasure_eq_measure)
hoelzl@50419:   next
hoelzl@50419:     assume "b < t" 
hoelzl@50419:     have "1 = emeasure M {x\<in>space M. X x \<le> b}"
hoelzl@50419:       using distr[of b] `a < b` by (simp add: one_ereal_def emeasure_eq_measure)
hoelzl@50419:     also have "\<dots> \<le> emeasure M {x\<in>space M. X x \<le> t}"
hoelzl@50419:       using X `b < t` by (auto intro!: emeasure_mono measurable_sets)
hoelzl@50419:     finally have "emeasure M {x\<in>space M. X x \<le> t} = 1"
hoelzl@50419:        by (simp add: antisym emeasure_eq_measure one_ereal_def)
hoelzl@50419:     with `b < t` `a < b` show ?thesis by (simp add: measure_def one_ereal_def)
hoelzl@50419:   qed
hoelzl@50419: qed (insert X `a < b`, auto)
hoelzl@50419: 
hoelzl@50419: lemma (in prob_space) uniform_distributed_measure:
hoelzl@50419:   fixes a b :: real
hoelzl@50419:   assumes D: "distributed M lborel X (\<lambda>x. indicator {a .. b} x / measure lborel {a .. b})"
hoelzl@50419:   assumes " a \<le> t" "t \<le> b"
hoelzl@50419:   shows "\<P>(x in M. X x \<le> t) = (t - a) / (b - a)"
hoelzl@50419: proof -
hoelzl@50419:   have "emeasure M {x \<in> space M. X x \<le> t} = emeasure (distr M lborel X) {.. t}"
hoelzl@50419:     using distributed_measurable[OF D]
hoelzl@50419:     by (subst emeasure_distr) (auto intro!: arg_cong2[where f=emeasure])
wenzelm@53015:   also have "\<dots> = (\<integral>\<^sup>+x. ereal (1 / (b - a)) * indicator {a .. t} x \<partial>lborel)"
hoelzl@50419:     using distributed_borel_measurable[OF D] `a \<le> t` `t \<le> b`
hoelzl@50419:     unfolding distributed_distr_eq_density[OF D]
hoelzl@50419:     by (subst emeasure_density)
hoelzl@56996:        (auto intro!: nn_integral_cong simp: measure_def split: split_indicator)
hoelzl@50419:   also have "\<dots> = ereal (1 / (b - a)) * (t - a)"
hoelzl@50419:     using `a \<le> t` `t \<le> b`
hoelzl@56996:     by (subst nn_integral_cmult_indicator) auto
hoelzl@50419:   finally show ?thesis
hoelzl@50419:     by (simp add: measure_def)
hoelzl@50419: qed
hoelzl@50419: 
hoelzl@50419: lemma (in prob_space) uniform_distributed_bounds:
hoelzl@50419:   fixes a b :: real
hoelzl@50419:   assumes D: "distributed M lborel X (\<lambda>x. indicator {a .. b} x / measure lborel {a .. b})"
hoelzl@50419:   shows "a < b"
hoelzl@50419: proof (rule ccontr)
hoelzl@50419:   assume "\<not> a < b"
hoelzl@50419:   then have "{a .. b} = {} \<or> {a .. b} = {a .. a}" by simp
hoelzl@50419:   with uniform_distributed_params[OF D] show False 
hoelzl@50419:     by (auto simp: measure_def)
hoelzl@50419: qed
hoelzl@50419: 
hoelzl@50419: lemma (in prob_space) uniform_distributed_iff:
hoelzl@50419:   fixes a b :: real
hoelzl@50419:   shows "distributed M lborel X (\<lambda>x. indicator {a..b} x / measure lborel {a..b}) \<longleftrightarrow> 
hoelzl@50419:     (X \<in> borel_measurable M \<and> a < b \<and> (\<forall>t\<in>{a .. b}. \<P>(x in M. X x \<le> t)= (t - a) / (b - a)))"
hoelzl@50419:   using
hoelzl@50419:     uniform_distributed_bounds[of X a b]
hoelzl@50419:     uniform_distributed_measure[of X a b]
hoelzl@50419:     distributed_measurable[of M lborel X]
hoelzl@57447:   by (auto intro!: uniform_distrI_borel_atLeastAtMost)
hoelzl@50419: 
hoelzl@50419: lemma (in prob_space) uniform_distributed_expectation:
hoelzl@50419:   fixes a b :: real
hoelzl@50419:   assumes D: "distributed M lborel X (\<lambda>x. indicator {a .. b} x / measure lborel {a .. b})"
hoelzl@50419:   shows "expectation X = (a + b) / 2"
hoelzl@50419: proof (subst distributed_integral[OF D, of "\<lambda>x. x", symmetric])
hoelzl@50419:   have "a < b"
hoelzl@50419:     using uniform_distributed_bounds[OF D] .
hoelzl@50419: 
hoelzl@50419:   have "(\<integral> x. indicator {a .. b} x / measure lborel {a .. b} * x \<partial>lborel) = 
hoelzl@50419:     (\<integral> x. (x / measure lborel {a .. b}) * indicator {a .. b} x \<partial>lborel)"
hoelzl@50419:     by (intro integral_cong) auto
hoelzl@50419:   also have "(\<integral> x. (x / measure lborel {a .. b}) * indicator {a .. b} x \<partial>lborel) = (a + b) / 2"
hoelzl@57447:   proof (subst integral_FTC_Icc_real)
hoelzl@50419:     fix x
wenzelm@53077:     show "DERIV (\<lambda>x. x\<^sup>2 / (2 * measure lborel {a..b})) x :> x / measure lborel {a..b}"
hoelzl@50419:       using uniform_distributed_params[OF D]
hoelzl@56381:       by (auto intro!: derivative_eq_intros)
hoelzl@50419:     show "isCont (\<lambda>x. x / Sigma_Algebra.measure lborel {a..b}) x"
hoelzl@50419:       using uniform_distributed_params[OF D]
hoelzl@50419:       by (auto intro!: isCont_divide)
wenzelm@53015:     have *: "b\<^sup>2 / (2 * measure lborel {a..b}) - a\<^sup>2 / (2 * measure lborel {a..b}) =
hoelzl@50419:       (b*b - a * a) / (2 * (b - a))"
hoelzl@50419:       using `a < b`
hoelzl@50419:       by (auto simp: measure_def power2_eq_square diff_divide_distrib[symmetric])
wenzelm@53015:     show "b\<^sup>2 / (2 * measure lborel {a..b}) - a\<^sup>2 / (2 * measure lborel {a..b}) = (a + b) / 2"
hoelzl@50419:       using `a < b`
hoelzl@50419:       unfolding * square_diff_square_factored by (auto simp: field_simps)
hoelzl@50419:   qed (insert `a < b`, simp)
hoelzl@50419:   finally show "(\<integral> x. indicator {a .. b} x / measure lborel {a .. b} * x \<partial>lborel) = (a + b) / 2" .
hoelzl@50419: qed auto
hoelzl@50419: 
hoelzl@57235: lemma (in prob_space) uniform_distributed_variance:
hoelzl@57235:   fixes a b :: real
hoelzl@57235:   assumes D: "distributed M lborel X (\<lambda>x. indicator {a .. b} x / measure lborel {a .. b})"
hoelzl@57235:   shows "variance X = (b - a)\<^sup>2 / 12"
hoelzl@57235: proof (subst distributed_variance)
hoelzl@57235:   have [arith]: "a < b" using uniform_distributed_bounds[OF D] .
hoelzl@57235:   let ?\<mu> = "expectation X" let ?D = "\<lambda>x. indicator {a..b} (x + ?\<mu>) / measure lborel {a..b}"
hoelzl@57235:   have "(\<integral>x. x\<^sup>2 * (?D x) \<partial>lborel) = (\<integral>x. x\<^sup>2 * (indicator {a - ?\<mu> .. b - ?\<mu>} x) / measure lborel {a .. b} \<partial>lborel)"
hoelzl@57235:     by (intro integral_cong) (auto split: split_indicator)
hoelzl@57235:   also have "\<dots> = (b - a)\<^sup>2 / 12"
hoelzl@57447:     by (simp add: integral_power uniform_distributed_expectation[OF D])
hoelzl@57235:        (simp add: eval_nat_numeral field_simps )
hoelzl@57235:   finally show "(\<integral>x. x\<^sup>2 * ?D x \<partial>lborel) = (b - a)\<^sup>2 / 12" .
hoelzl@57235: qed fact
hoelzl@57235: 
hoelzl@57252: subsection {* Normal distribution *}
hoelzl@57252: 
hoelzl@57254: 
hoelzl@57252: definition normal_density :: "real \<Rightarrow> real \<Rightarrow> real \<Rightarrow> real" where
hoelzl@57252:   "normal_density \<mu> \<sigma> x = 1 / sqrt (2 * pi * \<sigma>\<^sup>2) * exp (-(x - \<mu>)\<^sup>2/ (2 * \<sigma>\<^sup>2))"
hoelzl@57252: 
hoelzl@57252: abbreviation std_normal_density :: "real \<Rightarrow> real" where
hoelzl@57252:   "std_normal_density \<equiv> normal_density 0 1"
hoelzl@57252: 
hoelzl@57252: lemma std_normal_density_def: "std_normal_density x = (1 / sqrt (2 * pi)) * exp (- x\<^sup>2 / 2)"
hoelzl@57252:   unfolding normal_density_def by simp
hoelzl@57252: 
hoelzl@57254: lemma normal_density_nonneg: "0 \<le> normal_density \<mu> \<sigma> x"
hoelzl@57254:   by (auto simp: normal_density_def)
hoelzl@57254: 
hoelzl@57254: lemma normal_density_pos: "0 < \<sigma> \<Longrightarrow> 0 < normal_density \<mu> \<sigma> x"
hoelzl@57254:   by (auto simp: normal_density_def)
hoelzl@57254: 
hoelzl@57252: lemma borel_measurable_normal_density[measurable]: "normal_density \<mu> \<sigma> \<in> borel_measurable borel"
hoelzl@57252:   by (auto simp: normal_density_def[abs_def])
hoelzl@57252: 
hoelzl@57254: lemma gaussian_moment_0:
hoelzl@57254:   "has_bochner_integral lborel (\<lambda>x. indicator {0..} x *\<^sub>R exp (- x\<^sup>2)) (sqrt pi / 2)"
hoelzl@57252: proof -
hoelzl@57252:   let ?pI = "\<lambda>f. (\<integral>\<^sup>+s. f (s::real) * indicator {0..} s \<partial>lborel)"
hoelzl@57252:   let ?gauss = "\<lambda>x. exp (- x\<^sup>2)"
hoelzl@57252: 
hoelzl@57254:   let ?I = "indicator {0<..} :: real \<Rightarrow> real"
hoelzl@57254:   let ?ff= "\<lambda>x s. x * exp (- x\<^sup>2 * (1 + s\<^sup>2)) :: real"
hoelzl@57254: 
hoelzl@57254:   have *: "?pI ?gauss = (\<integral>\<^sup>+x. ?gauss x * ?I x \<partial>lborel)"
hoelzl@57254:     by (intro nn_integral_cong_AE AE_I[where N="{0}"]) (auto split: split_indicator)
hoelzl@57254: 
hoelzl@57254:   have "?pI ?gauss * ?pI ?gauss = (\<integral>\<^sup>+x. \<integral>\<^sup>+s. ?gauss x * ?gauss s * ?I s * ?I x \<partial>lborel \<partial>lborel)"
hoelzl@57254:     by (auto simp: nn_integral_nonneg nn_integral_cmult[symmetric] nn_integral_multc[symmetric] *
hoelzl@57254:              intro!: nn_integral_cong split: split_indicator)
hoelzl@57254:   also have "\<dots> = (\<integral>\<^sup>+x. \<integral>\<^sup>+s. ?ff x s * ?I s * ?I x \<partial>lborel \<partial>lborel)"
hoelzl@57254:   proof (rule nn_integral_cong, cases)
hoelzl@57254:     fix x :: real assume "x \<noteq> 0"
hoelzl@57254:     then show "(\<integral>\<^sup>+s. ?gauss x * ?gauss s * ?I s * ?I x \<partial>lborel) = (\<integral>\<^sup>+s. ?ff x s * ?I s * ?I x \<partial>lborel)"
hoelzl@57254:       by (subst nn_integral_real_affine[where t="0" and c="x"])
hoelzl@57254:          (auto simp: mult_exp_exp nn_integral_cmult[symmetric] field_simps zero_less_mult_iff
hoelzl@57254:                intro!: nn_integral_cong split: split_indicator)
hoelzl@57254:   qed simp
hoelzl@57254:   also have "... = \<integral>\<^sup>+s. \<integral>\<^sup>+x. ?ff x s * ?I s * ?I x \<partial>lborel \<partial>lborel"
hoelzl@57254:     by (rule lborel_pair.Fubini'[symmetric]) auto
hoelzl@57254:   also have "... = ?pI (\<lambda>s. ?pI (\<lambda>x. ?ff x s))"
hoelzl@57254:     by (rule nn_integral_cong_AE)
hoelzl@57254:        (auto intro!: nn_integral_cong_AE AE_I[where N="{0}"] split: split_indicator_asm)
hoelzl@57252:   also have "\<dots> = ?pI (\<lambda>s. ereal (1 / (2 * (1 + s\<^sup>2))))"
hoelzl@57252:   proof (intro nn_integral_cong ereal_right_mult_cong)
hoelzl@57254:     fix s :: real show "?pI (\<lambda>x. ?ff x s) = ereal (1 / (2 * (1 + s\<^sup>2)))"
hoelzl@57252:     proof (subst nn_integral_FTC_atLeast)
hoelzl@57252:       have "((\<lambda>a. - (exp (- (a\<^sup>2 * (1 + s\<^sup>2))) / (2 + 2 * s\<^sup>2))) ---> (- (0 / (2 + 2 * s\<^sup>2)))) at_top"
hoelzl@57252:         apply (intro tendsto_intros filterlim_compose[OF exp_at_bot] filterlim_compose[OF filterlim_uminus_at_bot_at_top])
haftmann@57512:         apply (subst mult.commute)         
hoelzl@57254:         apply (auto intro!: filterlim_tendsto_pos_mult_at_top
hoelzl@57254:                             filterlim_at_top_mult_at_top[OF filterlim_ident filterlim_ident] 
hoelzl@57254:                     simp: add_pos_nonneg  power2_eq_square add_nonneg_eq_0_iff)
hoelzl@57254:         done
hoelzl@57252:       then show "((\<lambda>a. - (exp (- a\<^sup>2 - s\<^sup>2 * a\<^sup>2) / (2 + 2 * s\<^sup>2))) ---> 0) at_top"
hoelzl@57252:         by (simp add: field_simps)
hoelzl@57252:     qed (auto intro!: derivative_eq_intros simp: field_simps add_nonneg_eq_0_iff)
hoelzl@57252:   qed
hoelzl@57252:   also have "... = ereal (pi / 4)"
hoelzl@57252:   proof (subst nn_integral_FTC_atLeast)
hoelzl@57252:     show "((\<lambda>a. arctan a / 2) ---> (pi / 2) / 2 ) at_top"
hoelzl@57252:       by (intro tendsto_intros) (simp_all add: tendsto_arctan_at_top)
hoelzl@57252:   qed (auto intro!: derivative_eq_intros simp: add_nonneg_eq_0_iff field_simps power2_eq_square)
hoelzl@57252:   finally have "?pI ?gauss^2 = pi / 4"
hoelzl@57252:     by (simp add: power2_eq_square)
hoelzl@57252:   then have "?pI ?gauss = sqrt (pi / 4)"
hoelzl@57252:     using power_eq_iff_eq_base[of 2 "real (?pI ?gauss)" "sqrt (pi / 4)"]
hoelzl@57254:           nn_integral_nonneg[of lborel "\<lambda>x. ?gauss x * indicator {0..} x"]
hoelzl@57252:     by (cases "?pI ?gauss") auto
hoelzl@57254:   also have "?pI ?gauss = (\<integral>\<^sup>+x. indicator {0..} x *\<^sub>R exp (- x\<^sup>2) \<partial>lborel)"
hoelzl@57254:     by (intro nn_integral_cong) (simp split: split_indicator)
hoelzl@57254:   also have "sqrt (pi / 4) = sqrt pi / 2"
hoelzl@57254:     by (simp add: real_sqrt_divide)
hoelzl@57254:   finally show ?thesis
hoelzl@57254:     by (rule has_bochner_integral_nn_integral[rotated 2]) auto
hoelzl@57254: qed
hoelzl@57254: 
hoelzl@57254: lemma gaussian_moment_1:
hoelzl@57254:   "has_bochner_integral lborel (\<lambda>x::real. indicator {0..} x *\<^sub>R (exp (- x\<^sup>2) * x)) (1 / 2)" 
hoelzl@57254: proof - 
hoelzl@57254:   have "(\<integral>\<^sup>+x. indicator {0..} x *\<^sub>R (exp (- x\<^sup>2) * x) \<partial>lborel) =
hoelzl@57254:     (\<integral>\<^sup>+x. ereal (x * exp (- x\<^sup>2)) * indicator {0..} x \<partial>lborel)"
hoelzl@57254:     by (intro nn_integral_cong)
hoelzl@57254:        (auto simp: ac_simps times_ereal.simps(1)[symmetric] ereal_indicator simp del: times_ereal.simps)
hoelzl@57254:   also have "\<dots> = ereal (0 - (- exp (- 0\<^sup>2) / 2))"
hoelzl@57254:   proof (rule nn_integral_FTC_atLeast)
hoelzl@57254:     have "((\<lambda>x::real. - exp (- x\<^sup>2) / 2) ---> - 0 / 2) at_top"
hoelzl@57254:       by (intro tendsto_divide tendsto_minus filterlim_compose[OF exp_at_bot]
hoelzl@57275:                    filterlim_compose[OF filterlim_uminus_at_bot_at_top]
hoelzl@57275:                    filterlim_pow_at_top filterlim_ident)
hoelzl@57254:          auto
hoelzl@57254:     then show "((\<lambda>a::real. - exp (- a\<^sup>2) / 2) ---> 0) at_top"
hoelzl@57254:       by simp
hoelzl@57254:   qed (auto intro!: derivative_eq_intros)
hoelzl@57254:   also have "\<dots> = ereal (1 / 2)"
hoelzl@57254:     by simp
hoelzl@57254:   finally show ?thesis
hoelzl@57254:     by (rule has_bochner_integral_nn_integral[rotated 2]) (auto split: split_indicator)
hoelzl@57252: qed
hoelzl@57252: 
hoelzl@57254: lemma
hoelzl@57254:   fixes k :: nat
hoelzl@57254:   shows gaussian_moment_even_pos:
hoelzl@57254:     "has_bochner_integral lborel (\<lambda>x. indicator {0..} x *\<^sub>R (exp (-x\<^sup>2)*x^(2 * k)))
hoelzl@57254:        ((sqrt pi / 2) * (fact (2 * k) / (2 ^ (2 * k) * fact k)))" (is "?even")
hoelzl@57254:     and gaussian_moment_odd_pos:
hoelzl@57254:       "has_bochner_integral lborel (\<lambda>x. indicator {0..} x *\<^sub>R (exp (-x\<^sup>2)*x^(2 * k + 1))) (fact k / 2)" (is "?odd")
hoelzl@57254: proof -
hoelzl@57254:   let ?M = "\<lambda>k x. exp (- x\<^sup>2) * x^k :: real"
hoelzl@57254: 
hoelzl@57254:   { fix k I assume Mk: "has_bochner_integral lborel (\<lambda>x. indicator {0..} x *\<^sub>R ?M k x) I"
hoelzl@57254:     have "2 \<noteq> (0::real)"
hoelzl@57254:       by linarith
hoelzl@57254:     let ?f = "\<lambda>b. \<integral>x. indicator {0..} x *\<^sub>R ?M (k + 2) x * indicator {..b} x \<partial>lborel"
hoelzl@57254:     have "((\<lambda>b. (k + 1) / 2 * (\<integral>x. indicator {..b} x *\<^sub>R (indicator {0..} x *\<^sub>R ?M k x) \<partial>lborel) - ?M (k + 1) b / 2) --->
hoelzl@57254:         (k + 1) / 2 * (\<integral>x. indicator {0..} x *\<^sub>R ?M k x \<partial>lborel) - 0 / 2) at_top" (is ?tendsto)
hoelzl@57254:     proof (intro tendsto_intros `2 \<noteq> 0` tendsto_integral_at_top sets_lborel Mk[THEN integrable.intros])
hoelzl@57254:       show "(?M (k + 1) ---> 0) at_top"
hoelzl@57254:       proof cases
hoelzl@57254:         assume "even k"
hoelzl@57254:         have "((\<lambda>x. ((x\<^sup>2)^(k div 2 + 1) / exp (x\<^sup>2)) * (1 / x) :: real) ---> 0 * 0) at_top"
hoelzl@57254:           by (intro tendsto_intros tendsto_divide_0[OF tendsto_const] filterlim_compose[OF tendsto_power_div_exp_0]
hoelzl@57275:                    filterlim_at_top_imp_at_infinity filterlim_ident filterlim_pow_at_top filterlim_ident)
hoelzl@57275:              auto
hoelzl@57254:         also have "(\<lambda>x. ((x\<^sup>2)^(k div 2 + 1) / exp (x\<^sup>2)) * (1 / x) :: real) = ?M (k + 1)"
hoelzl@57254:           using `even k` by (auto simp: even_mult_two_ex fun_eq_iff exp_minus field_simps power2_eq_square power_mult)
hoelzl@57254:         finally show ?thesis by simp
hoelzl@57254:       next
hoelzl@57254:         assume "odd k"
hoelzl@57254:         have "((\<lambda>x. ((x\<^sup>2)^((k - 1) div 2 + 1) / exp (x\<^sup>2)) :: real) ---> 0) at_top"
hoelzl@57275:           by (intro filterlim_compose[OF tendsto_power_div_exp_0] filterlim_at_top_imp_at_infinity
hoelzl@57275:                     filterlim_ident filterlim_pow_at_top)
hoelzl@57275:              auto
hoelzl@57254:         also have "(\<lambda>x. ((x\<^sup>2)^((k - 1) div 2 + 1) / exp (x\<^sup>2)) :: real) = ?M (k + 1)"
hoelzl@57254:           using `odd k` by (auto simp: odd_Suc_mult_two_ex fun_eq_iff exp_minus field_simps power2_eq_square power_mult)
hoelzl@57254:         finally show ?thesis by simp
hoelzl@57254:       qed
hoelzl@57254:     qed
hoelzl@57254:     also have "?tendsto \<longleftrightarrow> ((?f ---> (k + 1) / 2 * (\<integral>x. indicator {0..} x *\<^sub>R ?M k x \<partial>lborel) - 0 / 2) at_top)"
hoelzl@57254:     proof (intro filterlim_cong refl eventually_at_top_linorder[THEN iffD2] exI[of _ 0] allI impI)
hoelzl@57254:       fix b :: real assume b: "0 \<le> b"
hoelzl@57254:       have "Suc k * (\<integral>x. indicator {0..b} x *\<^sub>R ?M k x \<partial>lborel) = (\<integral>x. indicator {0..b} x *\<^sub>R (exp (- x\<^sup>2) * ((Suc k) * x ^ k)) \<partial>lborel)"
hoelzl@57254:         unfolding integral_mult_right_zero[symmetric] by (intro integral_cong) auto
hoelzl@57254:       also have "\<dots> = exp (- b\<^sup>2) * b ^ (Suc k) - exp (- 0\<^sup>2) * 0 ^ (Suc k) -
hoelzl@57254:           (\<integral>x. indicator {0..b} x *\<^sub>R (- 2 * x * exp (- x\<^sup>2) * x ^ (Suc k)) \<partial>lborel)"
hoelzl@57254:         by (rule integral_by_parts')
hoelzl@57254:            (auto intro!: derivative_eq_intros b
hoelzl@57254:                  simp: real_of_nat_def[symmetric] diff_Suc real_of_nat_Suc field_simps split: nat.split)
hoelzl@57254:       also have "(\<integral>x. indicator {0..b} x *\<^sub>R (- 2 * x * exp (- x\<^sup>2) * x ^ (Suc k)) \<partial>lborel) =
hoelzl@57254:         (\<integral>x. indicator {0..b} x *\<^sub>R (- 2 * (exp (- x\<^sup>2) * x ^ (k + 2))) \<partial>lborel)"
hoelzl@57254:         by (intro integral_cong) auto
hoelzl@57254:       finally have "Suc k * (\<integral>x. indicator {0..b} x *\<^sub>R ?M k x \<partial>lborel) =
hoelzl@57254:         exp (- b\<^sup>2) * b ^ (Suc k) + 2 * (\<integral>x. indicator {0..b} x *\<^sub>R ?M (k + 2) x \<partial>lborel)"
hoelzl@57254:         apply (simp del: real_scaleR_def integral_mult_right add: integral_mult_right[symmetric])
hoelzl@57254:         apply (subst integral_mult_right_zero[symmetric])
hoelzl@57254:         apply (intro integral_cong)
hoelzl@57254:         apply simp_all
hoelzl@57254:         done
hoelzl@57254:       then show "(k + 1) / 2 * (\<integral>x. indicator {..b} x *\<^sub>R (indicator {0..} x *\<^sub>R ?M k x)\<partial>lborel) - exp (- b\<^sup>2) * b ^ (k + 1) / 2 = ?f b"
hoelzl@57254:         by (simp add: field_simps atLeastAtMost_def indicator_inter_arith)
hoelzl@57254:     qed
hoelzl@57254:     finally have int_M_at_top: "((?f ---> (k + 1) / 2 * (\<integral>x. indicator {0..} x *\<^sub>R ?M k x \<partial>lborel)) at_top)"
hoelzl@57254:       by simp
hoelzl@57254:     
hoelzl@57254:     have "has_bochner_integral lborel (\<lambda>x. indicator {0..} x *\<^sub>R ?M (k + 2) x) ((k + 1) / 2 * I)"
hoelzl@57254:     proof (rule has_bochner_integral_monotone_convergence_at_top)
hoelzl@57254:       fix y :: real
hoelzl@57254:       have *: "(\<lambda>x. indicator {0..} x *\<^sub>R ?M (k + 2) x * indicator {..y} x::real) =
hoelzl@57254:             (\<lambda>x. indicator {0..y} x *\<^sub>R ?M (k + 2) x)"
hoelzl@57254:         by rule (simp split: split_indicator)
hoelzl@57254:       show "integrable lborel (\<lambda>x. indicator {0..} x *\<^sub>R (?M (k + 2) x) * indicator {..y} x::real)"
hoelzl@57254:         unfolding * by (rule borel_integrable_compact) (auto intro!: continuous_intros)
hoelzl@57254:       show "((?f ---> (k + 1) / 2 * I) at_top)"
hoelzl@57254:         using int_M_at_top has_bochner_integral_integral_eq[OF Mk] by simp
hoelzl@57254:     qed (auto split: split_indicator) }
hoelzl@57254:   note step = this
hoelzl@57254: 
hoelzl@57254:   show ?even
hoelzl@57254:   proof (induct k)
hoelzl@57254:     case (Suc k)
hoelzl@57254:     note step[OF this]
hoelzl@57254:     also have "(real (2 * k + 1) / 2 * (sqrt pi / 2 * (real (fact (2 * k)) / real (2 ^ (2 * k) * fact k)))) =
hoelzl@57254:       sqrt pi / 2 * (real (fact (2 * Suc k)) / real (2 ^ (2 * Suc k) * fact (Suc k)))"
hoelzl@57254:       by (simp add: field_simps real_of_nat_Suc divide_simps del: fact_Suc) (simp add: field_simps)
hoelzl@57254:     finally show ?case
hoelzl@57254:       by simp
hoelzl@57254:   qed (insert gaussian_moment_0, simp)
hoelzl@57252: 
hoelzl@57254:   show ?odd
hoelzl@57254:   proof (induct k)
hoelzl@57254:     case (Suc k)
hoelzl@57254:     note step[OF this]
hoelzl@57254:     also have "(real (2 * k + 1 + 1) / 2 * (real (fact k) / 2)) = real (fact (Suc k)) / 2"
hoelzl@57254:       by (simp add: field_simps real_of_nat_Suc divide_simps del: fact_Suc) (simp add: field_simps)
hoelzl@57254:     finally show ?case
hoelzl@57254:       by simp
hoelzl@57254:   qed (insert gaussian_moment_1, simp)
hoelzl@57254: qed
hoelzl@57254: 
hoelzl@57254: context
hoelzl@57254:   fixes k :: nat and \<mu> \<sigma> :: real assumes [arith]: "0 < \<sigma>"
hoelzl@57254: begin
hoelzl@57254: 
hoelzl@57254: lemma normal_moment_even:
hoelzl@57254:   "has_bochner_integral lborel (\<lambda>x. normal_density \<mu> \<sigma> x * (x - \<mu>) ^ (2 * k)) (fact (2 * k) / ((2 / \<sigma>\<^sup>2)^k * fact k))"
hoelzl@57254: proof -
hoelzl@57254:   have eq: "\<And>x::real. x\<^sup>2^k = (x^k)\<^sup>2"
hoelzl@57254:     by (simp add: power_mult[symmetric] ac_simps)
hoelzl@57254: 
hoelzl@57254:   have "has_bochner_integral lborel (\<lambda>x. exp (-x\<^sup>2)*x^(2 * k))
hoelzl@57254:       (sqrt pi * (fact (2 * k) / (2 ^ (2 * k) * fact k)))"
hoelzl@57254:     using has_bochner_integral_even_function[OF gaussian_moment_even_pos[where k=k]] by simp
hoelzl@57254:   then have "has_bochner_integral lborel (\<lambda>x. (exp (-x\<^sup>2)*x^(2 * k)) * ((2*\<sigma>\<^sup>2)^k / sqrt (2 * pi * \<sigma>\<^sup>2)))
hoelzl@57254:       ((sqrt pi * (fact (2 * k) / (2 ^ (2 * k) * fact k))) * ((2*\<sigma>\<^sup>2)^k / sqrt (2 * pi * \<sigma>\<^sup>2)))"
hoelzl@57254:     by (rule has_bochner_integral_mult_left)
hoelzl@57254:   also have "(\<lambda>x. (exp (-x\<^sup>2)*x^(2 * k)) * ((2*\<sigma>\<^sup>2)^k / sqrt (2 * pi * \<sigma>\<^sup>2))) =
hoelzl@57254:     (\<lambda>x. exp (- ((sqrt 2 * \<sigma>) * x)\<^sup>2 / (2*\<sigma>\<^sup>2)) * ((sqrt 2 * \<sigma>) * x) ^ (2 * k) / sqrt (2 * pi * \<sigma>\<^sup>2))"
hoelzl@57254:     by (auto simp: fun_eq_iff field_simps real_sqrt_power[symmetric] real_sqrt_mult
hoelzl@57254:                    real_sqrt_divide power_mult eq)
hoelzl@57254:   also have "((sqrt pi * (fact (2 * k) / (2 ^ (2 * k) * fact k))) * ((2*\<sigma>\<^sup>2)^k / sqrt (2 * pi * \<sigma>\<^sup>2))) = 
hoelzl@57254:     (inverse (sqrt 2 * \<sigma>) * (real (fact (2 * k))) / ((2/\<sigma>\<^sup>2) ^ k * real (fact k)))"
hoelzl@57254:     by (auto simp: fun_eq_iff power_mult field_simps real_sqrt_power[symmetric] real_sqrt_mult
hoelzl@57254:                    power2_eq_square)
hoelzl@57254:   finally show ?thesis
hoelzl@57254:     unfolding normal_density_def
hoelzl@57254:     by (subst lborel_has_bochner_integral_real_affine_iff[where c="sqrt 2 * \<sigma>" and t=\<mu>]) simp_all
hoelzl@57254: qed
hoelzl@57252: 
hoelzl@57254: lemma normal_moment_abs_odd:
hoelzl@57254:   "has_bochner_integral lborel (\<lambda>x. normal_density \<mu> \<sigma> x * \<bar>x - \<mu>\<bar>^(2 * k + 1)) (2^k * \<sigma>^(2 * k + 1) * fact k * sqrt (2 / pi))"
hoelzl@57254: proof -
hoelzl@57254:   have "has_bochner_integral lborel (\<lambda>x. indicator {0..} x *\<^sub>R (exp (-x\<^sup>2)*\<bar>x\<bar>^(2 * k + 1))) (fact k / 2)"
hoelzl@57254:     by (rule has_bochner_integral_cong[THEN iffD1, OF _ _ _ gaussian_moment_odd_pos[of k]]) auto
hoelzl@57254:   from has_bochner_integral_even_function[OF this]
hoelzl@57254:   have "has_bochner_integral lborel (\<lambda>x. exp (-x\<^sup>2)*\<bar>x\<bar>^(2 * k + 1)) (fact k)"
hoelzl@57254:     by simp
hoelzl@57254:   then have "has_bochner_integral lborel (\<lambda>x. (exp (-x\<^sup>2)*\<bar>x\<bar>^(2 * k + 1)) * (2^k * \<sigma>^(2 * k + 1) / sqrt (pi * \<sigma>\<^sup>2)))
hoelzl@57254:       (fact k * (2^k * \<sigma>^(2 * k + 1) / sqrt (pi * \<sigma>\<^sup>2)))"
hoelzl@57254:     by (rule has_bochner_integral_mult_left)
hoelzl@57254:   also have "(\<lambda>x. (exp (-x\<^sup>2)*\<bar>x\<bar>^(2 * k + 1)) * (2^k * \<sigma>^(2 * k + 1) / sqrt (pi * \<sigma>\<^sup>2))) =
hoelzl@57254:     (\<lambda>x. exp (- (((sqrt 2 * \<sigma>) * x)\<^sup>2 / (2 * \<sigma>\<^sup>2))) * \<bar>sqrt 2 * \<sigma> * x\<bar> ^ (2 * k + 1) / sqrt (2 * pi * \<sigma>\<^sup>2))"
hoelzl@57254:     by (simp add: field_simps abs_mult real_sqrt_power[symmetric] power_mult real_sqrt_mult)
hoelzl@57254:   also have "(fact k * (2^k * \<sigma>^(2 * k + 1) / sqrt (pi * \<sigma>\<^sup>2))) = 
hoelzl@57254:     (inverse (sqrt 2) * inverse \<sigma> * (2 ^ k * (\<sigma> * \<sigma> ^ (2 * k)) * real (fact k) * sqrt (2 / pi)))"
hoelzl@57254:     by (auto simp: fun_eq_iff power_mult field_simps real_sqrt_power[symmetric] real_sqrt_divide
hoelzl@57254:                    real_sqrt_mult)
hoelzl@57254:   finally show ?thesis
hoelzl@57254:     unfolding normal_density_def
hoelzl@57254:     by (subst lborel_has_bochner_integral_real_affine_iff[where c="sqrt 2 * \<sigma>" and t=\<mu>])
hoelzl@57254:        simp_all
hoelzl@57254: qed
hoelzl@57254: 
hoelzl@57254: lemma normal_moment_odd:
hoelzl@57254:   "has_bochner_integral lborel (\<lambda>x. normal_density \<mu> \<sigma> x * (x - \<mu>)^(2 * k + 1)) 0"
hoelzl@57254: proof -
hoelzl@57254:   have "has_bochner_integral lborel (\<lambda>x. exp (- x\<^sup>2) * x^(2 * k + 1)::real) 0"
hoelzl@57254:     using gaussian_moment_odd_pos by (rule has_bochner_integral_odd_function) simp
hoelzl@57254:   then have "has_bochner_integral lborel (\<lambda>x. (exp (-x\<^sup>2)*x^(2 * k + 1)) * (2^k*\<sigma>^(2*k)/sqrt pi))
hoelzl@57254:       (0 * (2^k*\<sigma>^(2*k)/sqrt pi))"
hoelzl@57254:     by (rule has_bochner_integral_mult_left)
hoelzl@57254:   also have "(\<lambda>x. (exp (-x\<^sup>2)*x^(2 * k + 1)) * (2^k*\<sigma>^(2*k)/sqrt pi)) =
hoelzl@57254:     (\<lambda>x. exp (- ((sqrt 2 * \<sigma> * x)\<^sup>2 / (2 * \<sigma>\<^sup>2))) *
hoelzl@57254:           (sqrt 2 * \<sigma> * x * (sqrt 2 * \<sigma> * x) ^ (2 * k)) /
hoelzl@57254:           sqrt (2 * pi * \<sigma>\<^sup>2))"
hoelzl@57254:     unfolding real_sqrt_mult
hoelzl@57254:     by (simp add: field_simps abs_mult real_sqrt_power[symmetric] power_mult fun_eq_iff)
hoelzl@57254:   finally show ?thesis
hoelzl@57254:     unfolding normal_density_def
hoelzl@57254:     by (subst lborel_has_bochner_integral_real_affine_iff[where c="sqrt 2 * \<sigma>" and t=\<mu>]) simp_all
hoelzl@57254: qed
hoelzl@57254: 
hoelzl@57254: lemma integral_normal_moment_even:
hoelzl@57254:   "integral\<^sup>L lborel (\<lambda>x. normal_density \<mu> \<sigma> x * (x - \<mu>)^(2 * k)) = fact (2 * k) / ((2 / \<sigma>\<^sup>2)^k * fact k)"
hoelzl@57254:   using normal_moment_even by (rule has_bochner_integral_integral_eq)
hoelzl@57254: 
hoelzl@57254: lemma integral_normal_moment_abs_odd:
hoelzl@57254:   "integral\<^sup>L lborel (\<lambda>x. normal_density \<mu> \<sigma> x * \<bar>x - \<mu>\<bar>^(2 * k + 1)) = 2 ^ k * \<sigma> ^ (2 * k + 1) * fact k * sqrt (2 / pi)"
hoelzl@57254:   using normal_moment_abs_odd by (rule has_bochner_integral_integral_eq)
hoelzl@57254: 
hoelzl@57254: lemma integral_normal_moment_odd:
hoelzl@57254:   "integral\<^sup>L lborel (\<lambda>x. normal_density \<mu> \<sigma> x * (x - \<mu>)^(2 * k + 1)) = 0"
hoelzl@57254:   using normal_moment_odd by (rule has_bochner_integral_integral_eq)
hoelzl@57254: 
hoelzl@57254: end
hoelzl@57254: 
hoelzl@57252: 
hoelzl@57252: context
hoelzl@57252:   fixes \<sigma> :: real
hoelzl@57252:   assumes \<sigma>_pos[arith]: "0 < \<sigma>"
hoelzl@57252: begin
hoelzl@57252: 
hoelzl@57254: lemma normal_moment_nz_1: "has_bochner_integral lborel (\<lambda>x. normal_density \<mu> \<sigma> x * x) \<mu>"
hoelzl@57254: proof -
hoelzl@57254:   note normal_moment_even[OF \<sigma>_pos, of \<mu> 0]
hoelzl@57254:   note normal_moment_odd[OF \<sigma>_pos, of \<mu> 0] has_bochner_integral_mult_left[of \<mu>, OF this]
hoelzl@57254:   note has_bochner_integral_add[OF this]
hoelzl@57254:   then show ?thesis
hoelzl@57254:     by (simp add: power2_eq_square field_simps)  
hoelzl@57254: qed
hoelzl@57254: 
hoelzl@57254: lemma integral_normal_moment_nz_1:
hoelzl@57254:   "integral\<^sup>L lborel (\<lambda>x. normal_density \<mu> \<sigma> x * x) = \<mu>"
hoelzl@57254:   using normal_moment_nz_1 by (rule has_bochner_integral_integral_eq)
hoelzl@57254: 
hoelzl@57254: lemma integrable_normal_moment_nz_1: "integrable lborel (\<lambda>x. normal_density \<mu> \<sigma> x * x)"
hoelzl@57254:   using normal_moment_nz_1 by rule
hoelzl@57252: 
hoelzl@57254: lemma integrable_normal_moment: "integrable lborel (\<lambda>x. normal_density \<mu> \<sigma> x * (x - \<mu>)^k)"
hoelzl@57254: proof cases
hoelzl@57254:   assume "even k" then show ?thesis
hoelzl@57254:     using integrable.intros[OF normal_moment_even] by (auto simp add: even_mult_two_ex)
hoelzl@57254: next
hoelzl@57254:   assume "odd k" then show ?thesis
hoelzl@57254:     using integrable.intros[OF normal_moment_odd] by (auto simp add: odd_Suc_mult_two_ex)
hoelzl@57254: qed
hoelzl@57252: 
hoelzl@57254: lemma integrable_normal_moment_abs: "integrable lborel (\<lambda>x. normal_density \<mu> \<sigma> x * \<bar>x - \<mu>\<bar>^k)"
hoelzl@57254: proof cases
hoelzl@57254:   assume "even k" then show ?thesis
hoelzl@57254:     using integrable.intros[OF normal_moment_even] by (auto simp add: even_mult_two_ex power_even_abs)
hoelzl@57254: next
hoelzl@57254:   assume "odd k" then show ?thesis
hoelzl@57254:     using integrable.intros[OF normal_moment_abs_odd] by (auto simp add: odd_Suc_mult_two_ex)
hoelzl@57254: qed
hoelzl@57254: 
hoelzl@57254: lemma integrable_normal_density[simp, intro]: "integrable lborel (normal_density \<mu> \<sigma>)"
hoelzl@57254:   using integrable_normal_moment[of \<mu> 0] by simp
hoelzl@57252: 
hoelzl@57252: lemma integral_normal_density[simp]: "(\<integral>x. normal_density \<mu> \<sigma> x \<partial>lborel) = 1"
hoelzl@57254:   using integral_normal_moment_even[of \<sigma> \<mu> 0] by simp
hoelzl@57252: 
hoelzl@57252: lemma prob_space_normal_density:
hoelzl@57254:   "prob_space (density lborel (normal_density \<mu> \<sigma>))"
hoelzl@57254:   proof qed (simp add: emeasure_density nn_integral_eq_integral normal_density_nonneg)
hoelzl@57254: 
hoelzl@57254: end
hoelzl@57254: 
hoelzl@57254: 
hoelzl@57254: 
hoelzl@57254: context
hoelzl@57254:   fixes k :: nat
hoelzl@57254: begin
hoelzl@57254: 
hoelzl@57254: lemma std_normal_moment_even:
hoelzl@57254:   "has_bochner_integral lborel (\<lambda>x. std_normal_density x * x ^ (2 * k)) (fact (2 * k) / (2^k * fact k))"
hoelzl@57254:   using normal_moment_even[of 1 0 k] by simp
hoelzl@57254: 
hoelzl@57254: lemma std_normal_moment_abs_odd:
hoelzl@57254:   "has_bochner_integral lborel (\<lambda>x. std_normal_density x * \<bar>x\<bar>^(2 * k + 1)) (sqrt (2/pi) * 2^k * fact k)"
hoelzl@57254:   using normal_moment_abs_odd[of 1 0 k] by (simp add: ac_simps)
hoelzl@57254: 
hoelzl@57254: lemma std_normal_moment_odd:
hoelzl@57254:   "has_bochner_integral lborel (\<lambda>x. std_normal_density x * x^(2 * k + 1)) 0"
hoelzl@57254:   using normal_moment_odd[of 1 0 k] by simp
hoelzl@57254: 
hoelzl@57254: lemma integral_std_normal_moment_even:
hoelzl@57254:   "integral\<^sup>L lborel (\<lambda>x. std_normal_density x * x^(2*k)) = fact (2 * k) / (2^k * fact k)"
hoelzl@57254:   using std_normal_moment_even by (rule has_bochner_integral_integral_eq)
hoelzl@57254: 
hoelzl@57254: lemma integral_std_normal_moment_abs_odd:
hoelzl@57254:   "integral\<^sup>L lborel (\<lambda>x. std_normal_density x * \<bar>x\<bar>^(2 * k + 1)) = sqrt (2 / pi) * 2^k * fact k"
hoelzl@57254:   using std_normal_moment_abs_odd by (rule has_bochner_integral_integral_eq)
hoelzl@57254: 
hoelzl@57254: lemma integral_std_normal_moment_odd:
hoelzl@57254:   "integral\<^sup>L lborel (\<lambda>x. std_normal_density x * x^(2 * k + 1)) = 0"
hoelzl@57254:   using std_normal_moment_odd by (rule has_bochner_integral_integral_eq)
hoelzl@57254: 
hoelzl@57254: lemma integrable_std_normal_moment_abs: "integrable lborel (\<lambda>x. std_normal_density x * \<bar>x\<bar>^k)"
hoelzl@57254:   using integrable_normal_moment_abs[of 1 0 k] by simp
hoelzl@57254: 
hoelzl@57254: lemma integrable_std_normal_moment: "integrable lborel (\<lambda>x. std_normal_density x * x^k)"
hoelzl@57254:   using integrable_normal_moment[of 1 0 k] by simp
hoelzl@57252: 
hoelzl@50419: end
hoelzl@57252: 
hoelzl@57252: lemma (in prob_space) normal_density_affine:
hoelzl@57252:   assumes X: "distributed M lborel X (normal_density \<mu> \<sigma>)"
hoelzl@57252:   assumes [simp, arith]: "0 < \<sigma>" "\<alpha> \<noteq> 0"
hoelzl@57252:   shows "distributed M lborel (\<lambda>x. \<beta> + \<alpha> * X x) (normal_density (\<beta> + \<alpha> * \<mu>) (\<bar>\<alpha>\<bar> * \<sigma>))"
hoelzl@57252: proof -
hoelzl@57252:   have eq: "\<And>x. \<bar>\<alpha>\<bar> * normal_density (\<beta> + \<alpha> * \<mu>) (\<bar>\<alpha>\<bar> * \<sigma>) (x * \<alpha> + \<beta>) =
hoelzl@57252:     normal_density \<mu> \<sigma> x"
hoelzl@57252:     by (simp add: normal_density_def real_sqrt_mult field_simps)
hoelzl@57252:        (simp add: power2_eq_square field_simps)
hoelzl@57252:   show ?thesis
hoelzl@57252:     by (rule distributed_affineI[OF _ `\<alpha> \<noteq> 0`, where t=\<beta>]) (simp_all add: eq X)
hoelzl@57252: qed
hoelzl@57252: 
hoelzl@57252: lemma (in prob_space) normal_standard_normal_convert:
hoelzl@57252:   assumes pos_var[simp, arith]: "0 < \<sigma>"
hoelzl@57252:   shows "distributed M lborel X (normal_density  \<mu> \<sigma>) = distributed M lborel (\<lambda>x. (X x - \<mu>) / \<sigma>) std_normal_density"
hoelzl@57252: proof auto
hoelzl@57252:   assume "distributed M lborel X (\<lambda>x. ereal (normal_density \<mu> \<sigma> x))"
hoelzl@57252:   then have "distributed M lborel (\<lambda>x. -\<mu> / \<sigma> + (1/\<sigma>) * X x) (\<lambda>x. ereal (normal_density (-\<mu> / \<sigma> + (1/\<sigma>)* \<mu>) (\<bar>1/\<sigma>\<bar> * \<sigma>) x))"
hoelzl@57252:     by(rule normal_density_affine) auto
hoelzl@57252:   
hoelzl@57252:   then show "distributed M lborel (\<lambda>x. (X x - \<mu>) / \<sigma>) (\<lambda>x. ereal (std_normal_density x))"
hoelzl@57252:     by (simp add: diff_divide_distrib[symmetric] field_simps)
hoelzl@57252: next
hoelzl@57252:   assume *: "distributed M lborel (\<lambda>x. (X x - \<mu>) / \<sigma>) (\<lambda>x. ereal (std_normal_density x))"
hoelzl@57252:   have "distributed M lborel (\<lambda>x. \<mu> + \<sigma> * ((X x - \<mu>) / \<sigma>)) (\<lambda>x. ereal (normal_density \<mu> \<bar>\<sigma>\<bar> x))"
hoelzl@57252:     using normal_density_affine[OF *, of \<sigma> \<mu>] by simp  
hoelzl@57252:   then show "distributed M lborel X (\<lambda>x. ereal (normal_density \<mu> \<sigma> x))" by simp
hoelzl@57252: qed
hoelzl@57252: 
hoelzl@57252: lemma conv_normal_density_zero_mean:
hoelzl@57252:   assumes [simp, arith]: "0 < \<sigma>" "0 < \<tau>"
hoelzl@57252:   shows "(\<lambda>x. \<integral>\<^sup>+y. ereal (normal_density 0 \<sigma> (x - y) * normal_density 0 \<tau> y) \<partial>lborel) =
hoelzl@57252:     normal_density 0 (sqrt (\<sigma>\<^sup>2 + \<tau>\<^sup>2))"  (is "?LHS = ?RHS")
hoelzl@57252: proof -
hoelzl@57252:   def \<sigma>' \<equiv> "\<sigma>\<^sup>2" and \<tau>' \<equiv> "\<tau>\<^sup>2"
hoelzl@57252:   then have [simp, arith]: "0 < \<sigma>'" "0 < \<tau>'"
hoelzl@57252:     by simp_all
hoelzl@57252:   let ?\<sigma> = "sqrt ((\<sigma>' * \<tau>') / (\<sigma>' + \<tau>'))"  
hoelzl@57252:   have sqrt: "(sqrt (2 * pi * (\<sigma>' + \<tau>')) * sqrt (2 * pi * (\<sigma>' * \<tau>') / (\<sigma>' + \<tau>'))) = 
hoelzl@57252:     (sqrt (2 * pi * \<sigma>') * sqrt (2 * pi * \<tau>'))"
hoelzl@57252:     by (subst power_eq_iff_eq_base[symmetric, where n=2])
hoelzl@57252:        (simp_all add: real_sqrt_mult[symmetric] power2_eq_square)
hoelzl@57252:   have "?LHS =
hoelzl@57252:     (\<lambda>x. \<integral>\<^sup>+y. ereal((normal_density 0 (sqrt (\<sigma>' + \<tau>')) x) * normal_density (\<tau>' * x / (\<sigma>' + \<tau>')) ?\<sigma> y) \<partial>lborel)"
hoelzl@57252:     apply (intro ext nn_integral_cong)
hoelzl@57252:     apply (simp add: normal_density_def \<sigma>'_def[symmetric] \<tau>'_def[symmetric] sqrt mult_exp_exp)
hoelzl@57252:     apply (simp add: divide_simps power2_eq_square)
hoelzl@57252:     apply (simp add: field_simps)
hoelzl@57252:     done
hoelzl@57252: 
hoelzl@57252:   also have "... =
hoelzl@57252:     (\<lambda>x. (normal_density 0 (sqrt (\<sigma>\<^sup>2 + \<tau>\<^sup>2)) x) * \<integral>\<^sup>+y. ereal( normal_density (\<tau>\<^sup>2* x / (\<sigma>\<^sup>2 + \<tau>\<^sup>2)) ?\<sigma> y) \<partial>lborel)"
hoelzl@57252:     by (subst nn_integral_cmult[symmetric]) (auto simp: \<sigma>'_def \<tau>'_def normal_density_def)
hoelzl@57252: 
hoelzl@57252:   also have "... = (\<lambda>x. (normal_density 0 (sqrt (\<sigma>\<^sup>2 + \<tau>\<^sup>2)) x))"
hoelzl@57254:     by (subst nn_integral_eq_integral) (auto simp: normal_density_nonneg)
hoelzl@57252: 
hoelzl@57252:   finally show ?thesis by fast
hoelzl@57252: qed
hoelzl@57252: 
hoelzl@57252: lemma conv_std_normal_density:
hoelzl@57252:   "(\<lambda>x. \<integral>\<^sup>+y. ereal (std_normal_density (x - y) * std_normal_density y) \<partial>lborel) =
hoelzl@57252:   (normal_density 0 (sqrt 2))"
hoelzl@57252:   by (subst conv_normal_density_zero_mean) simp_all
hoelzl@57252: 
hoelzl@57252: lemma (in prob_space) sum_indep_normal:
hoelzl@57252:   assumes indep: "indep_var borel X borel Y"
hoelzl@57252:   assumes pos_var[arith]: "0 < \<sigma>" "0 < \<tau>"
hoelzl@57252:   assumes normalX[simp]: "distributed M lborel X (normal_density \<mu> \<sigma>)"
hoelzl@57252:   assumes normalY[simp]: "distributed M lborel Y (normal_density \<nu> \<tau>)"
hoelzl@57252:   shows "distributed M lborel (\<lambda>x. X x + Y x) (normal_density (\<mu> + \<nu>) (sqrt (\<sigma>\<^sup>2 + \<tau>\<^sup>2)))"
hoelzl@57252: proof -
hoelzl@57252:   have ind[simp]: "indep_var borel (\<lambda>x. - \<mu> + X x) borel (\<lambda>x. - \<nu> + Y x)"
hoelzl@57252:   proof -
hoelzl@57252:     have "indep_var borel ( (\<lambda>x. -\<mu> + x) o X) borel ((\<lambda>x. - \<nu> + x) o Y)"
hoelzl@57252:       by (auto intro!: indep_var_compose assms) 
hoelzl@57252:     then show ?thesis by (simp add: o_def)
hoelzl@57252:   qed
hoelzl@57252:   
hoelzl@57252:   have "distributed M lborel (\<lambda>x. -\<mu> + 1 * X x) (normal_density (- \<mu> + 1 * \<mu>) (\<bar>1\<bar> * \<sigma>))"
hoelzl@57252:     by(rule normal_density_affine[OF normalX pos_var(1), of 1 "-\<mu>"]) simp
hoelzl@57252:   then have 1[simp]: "distributed M lborel (\<lambda>x. - \<mu> +  X x) (normal_density 0 \<sigma>)" by simp
hoelzl@57252: 
hoelzl@57252:   have "distributed M lborel (\<lambda>x. -\<nu> + 1 * Y x) (normal_density (- \<nu> + 1 * \<nu>) (\<bar>1\<bar> * \<tau>))"
hoelzl@57252:     by(rule normal_density_affine[OF normalY pos_var(2), of 1 "-\<nu>"]) simp
hoelzl@57252:   then have 2[simp]: "distributed M lborel (\<lambda>x. - \<nu> +  Y x) (normal_density 0 \<tau>)" by simp
hoelzl@57252:   
hoelzl@57252:   have *: "distributed M lborel (\<lambda>x. (- \<mu> + X x) + (- \<nu> + Y x)) (\<lambda>x. ereal (normal_density 0 (sqrt (\<sigma>\<^sup>2 + \<tau>\<^sup>2)) x))"
hoelzl@57252:     using distributed_convolution[OF ind 1 2] conv_normal_density_zero_mean[OF pos_var] by simp
hoelzl@57252:   
hoelzl@57252:   have "distributed M lborel (\<lambda>x. \<mu> + \<nu> + 1 * (- \<mu> + X x + (- \<nu> + Y x)))
hoelzl@57252:         (\<lambda>x. ereal (normal_density (\<mu> + \<nu> + 1 * 0) (\<bar>1\<bar> * sqrt (\<sigma>\<^sup>2 + \<tau>\<^sup>2)) x))"
hoelzl@57252:     by (rule normal_density_affine[OF *, of 1 "\<mu> + \<nu>"]) (auto simp: add_pos_pos)
hoelzl@57252: 
hoelzl@57252:   then show ?thesis by auto
hoelzl@57252: qed
hoelzl@57252: 
hoelzl@57252: lemma (in prob_space) diff_indep_normal:
hoelzl@57252:   assumes indep[simp]: "indep_var borel X borel Y"
hoelzl@57252:   assumes [simp, arith]: "0 < \<sigma>" "0 < \<tau>"
hoelzl@57252:   assumes normalX[simp]: "distributed M lborel X (normal_density \<mu> \<sigma>)"
hoelzl@57252:   assumes normalY[simp]: "distributed M lborel Y (normal_density \<nu> \<tau>)"
hoelzl@57252:   shows "distributed M lborel (\<lambda>x. X x - Y x) (normal_density (\<mu> - \<nu>) (sqrt (\<sigma>\<^sup>2 + \<tau>\<^sup>2)))"
hoelzl@57252: proof -
hoelzl@57252:   have "distributed M lborel (\<lambda>x. 0 + - 1 * Y x) (\<lambda>x. ereal (normal_density (0 + - 1 * \<nu>) (\<bar>- 1\<bar> * \<tau>) x))" 
hoelzl@57252:     by(rule normal_density_affine, auto)
hoelzl@57252:   then have [simp]:"distributed M lborel (\<lambda>x. - Y x) (\<lambda>x. ereal (normal_density (- \<nu>) \<tau> x))" by simp
hoelzl@57252: 
hoelzl@57252:   have "distributed M lborel (\<lambda>x. X x + (- Y x)) (normal_density (\<mu> + - \<nu>) (sqrt (\<sigma>\<^sup>2 + \<tau>\<^sup>2)))"
hoelzl@57252:     apply (rule sum_indep_normal)
hoelzl@57252:     apply (rule indep_var_compose[unfolded comp_def, of borel _ borel _ "\<lambda>x. x" _ "\<lambda>x. - x"])
hoelzl@57252:     apply simp_all
hoelzl@57252:     done
hoelzl@57252:   then show ?thesis by simp
hoelzl@57252: qed
hoelzl@57252: 
hoelzl@57252: lemma (in prob_space) setsum_indep_normal:
hoelzl@57252:   assumes "finite I" "I \<noteq> {}" "indep_vars (\<lambda>i. borel) X I"
hoelzl@57252:   assumes "\<And>i. i \<in> I \<Longrightarrow> 0 < \<sigma> i"
hoelzl@57252:   assumes normal: "\<And>i. i \<in> I \<Longrightarrow> distributed M lborel (X i) (normal_density (\<mu> i) (\<sigma> i))"
hoelzl@57252:   shows "distributed M lborel (\<lambda>x. \<Sum>i\<in>I. X i x) (normal_density (\<Sum>i\<in>I. \<mu> i) (sqrt (\<Sum>i\<in>I. (\<sigma> i)\<^sup>2)))"
hoelzl@57252:   using assms 
hoelzl@57252: proof (induct I rule: finite_ne_induct)
hoelzl@57252:   case (singleton i) then show ?case by (simp add : abs_of_pos)
hoelzl@57252: next
hoelzl@57252:   case (insert i I)
hoelzl@57252:     then have 1: "distributed M lborel (\<lambda>x. (X i x) + (\<Sum>i\<in>I. X i x)) 
hoelzl@57252:                 (normal_density (\<mu> i  + setsum \<mu> I)  (sqrt ((\<sigma> i)\<^sup>2 + (sqrt (\<Sum>i\<in>I. (\<sigma> i)\<^sup>2))\<^sup>2)))"
hoelzl@57252:       apply (intro sum_indep_normal indep_vars_setsum insert real_sqrt_gt_zero)  
hoelzl@57252:       apply (auto intro: indep_vars_subset intro!: setsum_pos)
hoelzl@57252:       apply fastforce
hoelzl@57252:       done
hoelzl@57252:     have 2: "(\<lambda>x. (X i x)+ (\<Sum>i\<in>I. X i x)) = (\<lambda>x. (\<Sum>j\<in>insert i I. X j x))"
hoelzl@57252:           "\<mu> i + setsum \<mu> I = setsum \<mu> (insert i I)"
hoelzl@57252:       using insert by auto
hoelzl@57252:   
hoelzl@57252:     have 3: "(sqrt ((\<sigma> i)\<^sup>2 + (sqrt (\<Sum>i\<in>I. (\<sigma> i)\<^sup>2))\<^sup>2)) = (sqrt (\<Sum>i\<in>insert i I. (\<sigma> i)\<^sup>2))"
hoelzl@57252:       using insert by (simp add: setsum_nonneg)
hoelzl@57252:   
hoelzl@57252:     show ?case using 1 2 3 by simp  
hoelzl@57252: qed
hoelzl@57252: 
hoelzl@57252: lemma (in prob_space) standard_normal_distributed_expectation:
hoelzl@57254:   assumes D: "distributed M lborel X std_normal_density"
hoelzl@57252:   shows "expectation X = 0"
hoelzl@57254:   using integral_std_normal_moment_odd[of 0]
hoelzl@57252:   by (auto simp: distributed_integral[OF D, of "\<lambda>x. x", symmetric])
hoelzl@57252: 
hoelzl@57252: lemma (in prob_space) normal_distributed_expectation:
hoelzl@57254:   assumes \<sigma>[arith]: "0 < \<sigma>"
hoelzl@57252:   assumes D: "distributed M lborel X (normal_density \<mu> \<sigma>)"
hoelzl@57252:   shows "expectation X = \<mu>"
hoelzl@57254:   using integral_normal_moment_nz_1[OF \<sigma>, of \<mu>] distributed_integral[OF D, of "\<lambda>x. x", symmetric]
hoelzl@57254:   by (auto simp: field_simps)
hoelzl@57252: 
hoelzl@57252: lemma (in prob_space) normal_distributed_variance:
hoelzl@57252:   fixes a b :: real
hoelzl@57254:   assumes [simp, arith]: "0 < \<sigma>"
hoelzl@57252:   assumes D: "distributed M lborel X (normal_density \<mu> \<sigma>)"
hoelzl@57252:   shows "variance X = \<sigma>\<^sup>2"
hoelzl@57254:   using integral_normal_moment_even[of \<sigma> \<mu> 1]
hoelzl@57254:   by (subst distributed_integral[OF D, symmetric])
hoelzl@57254:      (simp_all add: eval_nat_numeral normal_distributed_expectation[OF assms])
hoelzl@57252: 
hoelzl@57254: lemma (in prob_space) standard_normal_distributed_variance:
hoelzl@57254:   "distributed M lborel X std_normal_density \<Longrightarrow> variance X = 1"
hoelzl@57254:   using normal_distributed_variance[of 1 X 0] by simp
hoelzl@57252: 
hoelzl@57252: lemma (in information_space) entropy_normal_density:
hoelzl@57252:   assumes [arith]: "0 < \<sigma>"
hoelzl@57252:   assumes D: "distributed M lborel X (normal_density \<mu> \<sigma>)"
hoelzl@57252:   shows "entropy b lborel X = log b (2 * pi * exp 1 * \<sigma>\<^sup>2) / 2"
hoelzl@57252: proof -
hoelzl@57252:   have "entropy b lborel X = - (\<integral> x. normal_density \<mu> \<sigma> x * log b (normal_density \<mu> \<sigma> x) \<partial>lborel)"
hoelzl@57252:     using D by (rule entropy_distr)
hoelzl@57252:   also have "\<dots> = - (\<integral> x. normal_density \<mu> \<sigma> x * (- ln (2 * pi * \<sigma>\<^sup>2) - (x - \<mu>)\<^sup>2 / \<sigma>\<^sup>2) / (2 * ln b) \<partial>lborel)"
hoelzl@57252:     by (intro arg_cong[where f="uminus"] integral_cong)
hoelzl@57252:        (auto simp: normal_density_def field_simps ln_mult log_def ln_div ln_sqrt)
hoelzl@57252:   also have "\<dots> = - (\<integral>x. - (normal_density \<mu> \<sigma> x * (ln (2 * pi * \<sigma>\<^sup>2)) + (normal_density \<mu> \<sigma> x * (x - \<mu>)\<^sup>2) / \<sigma>\<^sup>2) / (2 * ln b) \<partial>lborel)"
hoelzl@57252:     by (intro arg_cong[where f="uminus"] integral_cong) (auto simp: divide_simps field_simps)
hoelzl@57252:   also have "\<dots> = (\<integral>x. normal_density \<mu> \<sigma> x * (ln (2 * pi * \<sigma>\<^sup>2)) + (normal_density \<mu> \<sigma> x * (x - \<mu>)\<^sup>2) / \<sigma>\<^sup>2 \<partial>lborel) / (2 * ln b)"
hoelzl@57252:     by (simp del: minus_add_distrib)
hoelzl@57252:   also have "\<dots> = (ln (2 * pi * \<sigma>\<^sup>2) + 1) / (2 * ln b)"
hoelzl@57254:     using integral_normal_moment_even[of \<sigma> \<mu> 1] by (simp add: integrable_normal_moment fact_numeral)
hoelzl@57252:   also have "\<dots> = log b (2 * pi * exp 1 * \<sigma>\<^sup>2) / 2"
hoelzl@57252:     by (simp add: log_def field_simps ln_mult)
hoelzl@57252:   finally show ?thesis .
hoelzl@57252: qed
hoelzl@57252: 
hoelzl@57252: end




