src/HOL/Probability/Distributions.thy

+(*  Title:      HOL/Probability/Distributions.thy
+    Author:     Sudeep Kanav, TU München
+    Author:     Johannes Hölzl, TU München *)
+
+header {* Properties of Various Distributions *}
+
 theory Distributions
-  imports Probability_Measure
+  imports Probability_Measure Convolution
 begin
 
+lemma measure_lebesgue_Icc: "measure lebesgue {a .. b} = (if a \<le> b then b - a else 0)"
+  by (auto simp: measure_def)
+
+lemma integral_power:
+  "a \<le> b \<Longrightarrow> (\<integral>x. x^k * indicator {a..b} x \<partial>lborel) = (b^Suc k - a^Suc k) / Suc k"
+proof (subst integral_FTC_atLeastAtMost)
+  fix x show "DERIV (\<lambda>x. x^Suc k / Suc k) x :> x^k"
+    by (intro derivative_eq_intros) auto
+qed (auto simp: field_simps)
+
+lemma has_bochner_integral_nn_integral:
+  assumes "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
+  assumes "(\<integral>\<^sup>+x. f x \<partial>M) = ereal x"
+  shows "has_bochner_integral M f x"
+  unfolding has_bochner_integral_iff
+proof
+  show "integrable M f"
+    using assms by (rule integrableI_nn_integral_finite)
+qed (auto simp: assms integral_eq_nn_integral)
+
+lemma (in prob_space) distributed_AE2:
+  assumes [measurable]: "distributed M N X f" "Measurable.pred N P"
+  shows "(AE x in M. P (X x)) \<longleftrightarrow> (AE x in N. 0 < f x \<longrightarrow> P x)"
+proof -
+  have "(AE x in M. P (X x)) \<longleftrightarrow> (AE x in distr M N X. P x)"
+    by (simp add: AE_distr_iff)
+  also have "\<dots> \<longleftrightarrow> (AE x in density N f. P x)"
+    unfolding distributed_distr_eq_density[OF assms(1)] ..
+  also have "\<dots> \<longleftrightarrow>  (AE x in N. 0 < f x \<longrightarrow> P x)"
+    by (rule AE_density) simp
+  finally show ?thesis .
+qed
+
+subsection {* Erlang *}
+
+lemma nn_intergal_power_times_exp_Icc:
+  assumes [arith]: "0 \<le> a"
+  shows "(\<integral>\<^sup>+x. ereal (x^k * exp (-x)) * indicator {0 .. a} x \<partial>lborel) =
+    (1 - (\<Sum>n\<le>k. (a^n * exp (-a)) / fact n)) * fact k" (is "?I = _")
+proof -
+  let ?f = "\<lambda>k x. x^k * exp (-x) / fact k"
+  let ?F = "\<lambda>k x. - (\<Sum>n\<le>k. (x^n * exp (-x)) / fact n)"
+
+  have "?I * (inverse (fact k)) = 
+      (\<integral>\<^sup>+x. ereal (x^k * exp (-x)) * indicator {0 .. a} x * (inverse (fact k)) \<partial>lborel)"
+    by (intro nn_integral_multc[symmetric]) auto
+  also have "\<dots> = (\<integral>\<^sup>+x. ereal (?f k x) * indicator {0 .. a} x \<partial>lborel)"
+    by (intro nn_integral_cong) (simp add: field_simps)
+  also have "\<dots> = ereal (?F k a) - (?F k 0)"
+  proof (rule nn_integral_FTC_atLeastAtMost)
+    fix x assume "x \<in> {0..a}"
+    show "DERIV (?F k) x :> ?f k x"
+    proof(induction k)
+      case 0 show ?case by (auto intro!: derivative_eq_intros)
+    next
+      case (Suc k)
+      have "DERIV (\<lambda>x. ?F k x - (x^Suc k * exp (-x)) / fact (Suc k)) x :>
+        ?f k x - ((real (Suc k) - x) * x ^ k * exp (- x)) / real (fact (Suc k))"
+        by (intro DERIV_diff Suc)
+           (auto intro!: derivative_eq_intros simp del: fact_Suc power_Suc
+                 simp add: field_simps power_Suc[symmetric] real_of_nat_def[symmetric])
+      also have "(\<lambda>x. ?F k x - (x^Suc k * exp (-x)) / fact (Suc k)) = ?F (Suc k)"
+        by simp
+      also have "?f k x - ((real (Suc k) - x) * x ^ k * exp (- x)) / real (fact (Suc k)) = ?f (Suc k) x"
+        by (auto simp: field_simps simp del: fact_Suc)
+           (simp_all add: real_of_nat_Suc field_simps)
+      finally show ?case .
+    qed
+  qed auto
+  also have "\<dots> = ereal (1 - (\<Sum>n\<le>k. (a^n * exp (-a)) / fact n))"
+    by (auto simp: power_0_left if_distrib[where f="\<lambda>x. x / a" for a] setsum_cases)
+  finally show ?thesis
+    by (cases "?I") (auto simp: field_simps)
+qed
+
+lemma nn_intergal_power_times_exp_Ici:
+  shows "(\<integral>\<^sup>+x. ereal (x^k * exp (-x)) * indicator {0 ..} x \<partial>lborel) = fact k"
+proof (rule LIMSEQ_unique)
+  let ?X = "\<lambda>n. \<integral>\<^sup>+ x. ereal (x^k * exp (-x)) * indicator {0 .. real n} x \<partial>lborel"
+  show "?X ----> (\<integral>\<^sup>+x. ereal (x^k * exp (-x)) * indicator {0 ..} x \<partial>lborel)"
+    apply (intro nn_integral_LIMSEQ)
+    apply (auto simp: incseq_def le_fun_def eventually_sequentially
+                split: split_indicator intro!: Lim_eventually)
+    apply (metis natceiling_le_eq)
+    done
+
+  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"
+    by (intro tendsto_intros tendsto_power_div_exp_0) simp
+  then show "?X ----> fact k"
+    by (subst nn_intergal_power_times_exp_Icc)
+       (auto simp: exp_minus field_simps intro!: filterlim_compose[OF _ filterlim_real_sequentially])
+qed
+
+definition erlang_density :: "nat \<Rightarrow> real \<Rightarrow> real \<Rightarrow> real" where
+  "erlang_density k l x = (if x < 0 then 0 else (l^(Suc k) * x^k * exp (- l * x)) / fact k)"
+
+definition erlang_CDF ::  "nat \<Rightarrow> real \<Rightarrow> real \<Rightarrow> real" where
+  "erlang_CDF k l x = (if x < 0 then 0 else 1 - (\<Sum>n\<le>k. ((l * x)^n * exp (- l * x) / fact n)))"
+
+lemma erlang_density_nonneg: "0 \<le> l \<Longrightarrow> 0 \<le> erlang_density k l x"
+  by (simp add: erlang_density_def)
+
+lemma borel_measurable_erlang_density[measurable]: "erlang_density k l \<in> borel_measurable borel"
+  by (auto simp add: erlang_density_def[abs_def])
+
+lemma erlang_CDF_transform: "0 < l \<Longrightarrow> erlang_CDF k l a = erlang_CDF k 1 (l * a)"
+  by (auto simp add: erlang_CDF_def mult_less_0_iff)
+
+lemma nn_integral_erlang_density:
+  assumes [arith]: "0 < l"
+  shows "(\<integral>\<^sup>+ x. ereal (erlang_density k l x) * indicator {.. a} x \<partial>lborel) = erlang_CDF k l a"
+proof cases
+  assume [arith]: "0 \<le> a"
+  have eq: "\<And>x. indicator {0..a} (x / l) = indicator {0..a*l} x"
+    by (simp add: field_simps split: split_indicator)
+  have "(\<integral>\<^sup>+x. ereal (erlang_density k l x) * indicator {.. a} x \<partial>lborel) =
+    (\<integral>\<^sup>+x. (l/fact k) * (ereal ((l*x)^k * exp (- (l*x))) * indicator {0 .. a} x) \<partial>lborel)"
+    by (intro nn_integral_cong) (auto simp: erlang_density_def power_mult_distrib split: split_indicator)
+  also have "\<dots> = (l/fact k) * (\<integral>\<^sup>+x. ereal ((l*x)^k * exp (- (l*x))) * indicator {0 .. a} x \<partial>lborel)"
+    by (intro nn_integral_cmult) auto
+  also have "\<dots> = ereal (l/fact k) * ((1/l) * (\<integral>\<^sup>+x. ereal (x^k * exp (- x)) * indicator {0 .. l * a} x \<partial>lborel))"
+    by (subst nn_integral_real_affine[where c="1 / l" and t=0]) (auto simp: field_simps eq)
+  also have "\<dots> = (1 - (\<Sum>n\<le>k. ((l * a)^n * exp (-(l * a))) / fact n))"
+    by (subst nn_intergal_power_times_exp_Icc) auto
+  also have "\<dots> = erlang_CDF k l a"
+    by (auto simp: erlang_CDF_def)
+  finally show ?thesis .
+next
+  assume "\<not> 0 \<le> a" 
+  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))"
+    by (intro nn_integral_cong) (auto simp: erlang_density_def)
+  ultimately show ?thesis
+    by (simp add: erlang_CDF_def)
+qed
+
+lemma emeasure_erlang_density: 
+  "0 < l \<Longrightarrow> emeasure (density lborel (erlang_density k l)) {.. a} = erlang_CDF k l a"
+  by (simp add: emeasure_density nn_integral_erlang_density)
+
+lemma nn_integral_erlang_ith_moment: 
+  fixes k i :: nat and l :: real
+  assumes [arith]: "0 < l" 
+  shows "(\<integral>\<^sup>+ x. ereal (erlang_density k l x * x ^ i) \<partial>lborel) = fact (k + i) / (fact k * l ^ i)"
+proof -
+  have eq: "\<And>x. indicator {0..} (x / l) = indicator {0..} x"
+    by (simp add: field_simps split: split_indicator)
+  have "(\<integral>\<^sup>+ x. ereal (erlang_density k l x * x^i) \<partial>lborel) =
+    (\<integral>\<^sup>+x. (l/(fact k * l^i)) * (ereal ((l*x)^(k+i) * exp (- (l*x))) * indicator {0 ..} x) \<partial>lborel)"
+    by (intro nn_integral_cong) (auto simp: erlang_density_def power_mult_distrib power_add split: split_indicator)
+  also have "\<dots> = (l/(fact k * l^i)) * (\<integral>\<^sup>+x. ereal ((l*x)^(k+i) * exp (- (l*x))) * indicator {0 ..} x \<partial>lborel)"
+    by (intro nn_integral_cmult) auto
+  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))"
+    by (subst nn_integral_real_affine[where c="1 / l" and t=0]) (auto simp: field_simps eq)
+  also have "\<dots> = fact (k + i) / (fact k * l ^ i)"
+    by (subst nn_intergal_power_times_exp_Ici) auto
+  finally show ?thesis .
+qed
+
+lemma prob_space_erlang_density:
+  assumes l[arith]: "0 < l"
+  shows "prob_space (density lborel (erlang_density k l))" (is "prob_space ?D")
+proof
+  show "emeasure ?D (space ?D) = 1"
+    using nn_integral_erlang_ith_moment[OF l, where k=k and i=0] by (simp add: emeasure_density)
+qed
+
+lemma (in prob_space) erlang_distributed_le:
+  assumes D: "distributed M lborel X (erlang_density k l)"
+  assumes [simp, arith]: "0 < l" "0 \<le> a"
+  shows "\<P>(x in M. X x \<le> a) = erlang_CDF k l a"
+proof -
+  have "emeasure M {x \<in> space M. X x \<le> a } = emeasure (distr M lborel X) {.. a}"
+    using distributed_measurable[OF D]
+    by (subst emeasure_distr) (auto intro!: arg_cong2[where f=emeasure])
+  also have "\<dots> = emeasure (density lborel (erlang_density k l)) {.. a}"
+    unfolding distributed_distr_eq_density[OF D] ..
+  also have "\<dots> = erlang_CDF k l a"
+    by (auto intro!: emeasure_erlang_density)
+  finally show ?thesis
+    by (auto simp: measure_def)
+qed
+
+lemma (in prob_space) erlang_distributed_gt:
+  assumes D[simp]: "distributed M lborel X (erlang_density k l)"
+  assumes [arith]: "0 < l" "0 \<le> a"
+  shows "\<P>(x in M. a < X x ) = 1 - (erlang_CDF k l a)"
+proof -
+  have " 1 - (erlang_CDF k l a) = 1 - \<P>(x in M. X x \<le> a)" by (subst erlang_distributed_le) auto
+  also have "\<dots> = prob (space M - {x \<in> space M. X x \<le> a })"
+    using distributed_measurable[OF D] by (auto simp: prob_compl)
+  also have "\<dots> = \<P>(x in M. a < X x )" by (auto intro!: arg_cong[where f=prob] simp: not_le)
+  finally show ?thesis by simp
+qed
+
+lemma erlang_CDF_at0: "erlang_CDF k l 0 = 0"
+  by (induction k) (auto simp: erlang_CDF_def)
+
+lemma erlang_distributedI:
+  assumes X[measurable]: "X \<in> borel_measurable M" and [arith]: "0 < l"
+    and X_distr: "\<And>a. 0 \<le> a \<Longrightarrow> emeasure M {x\<in>space M. X x \<le> a} = erlang_CDF k l a"
+  shows "distributed M lborel X (erlang_density k l)"
+proof (rule distributedI_borel_atMost)
+  fix a :: real
+  { assume "a \<le> 0"  
+    with X have "emeasure M {x\<in>space M. X x \<le> a} \<le> emeasure M {x\<in>space M. X x \<le> 0}"
+      by (intro emeasure_mono) auto
+    also have "... = 0"  by (auto intro!: erlang_CDF_at0 simp: X_distr[of 0])
+    finally have "emeasure M {x\<in>space M. X x \<le> a} \<le> 0" by simp
+    then have "emeasure M {x\<in>space M. X x \<le> a} = 0" by (simp add:emeasure_le_0_iff)
+  }
+  note eq_0 = this
+
+  show "(\<integral>\<^sup>+ x. erlang_density k l x * indicator {..a} x \<partial>lborel) = ereal (erlang_CDF k l a)"
+    using nn_integral_erlang_density[of l k a]
+    by (simp add: times_ereal.simps(1)[symmetric] ereal_indicator del: times_ereal.simps)
+
+  show "emeasure M {x\<in>space M. X x \<le> a} = ereal (erlang_CDF k l a)"
+    using X_distr[of a] eq_0 by (auto simp: one_ereal_def erlang_CDF_def)
+qed (simp_all add: erlang_density_nonneg)
+
+lemma (in prob_space) erlang_distributed_iff:
+  assumes [arith]: "0<l"
+  shows "distributed M lborel X (erlang_density k l) \<longleftrightarrow>
+    (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 ))"
+  using
+    distributed_measurable[of M lborel X "erlang_density k l"]
+    emeasure_erlang_density[of l]
+    erlang_distributed_le[of X k l]
+  by (auto intro!: erlang_distributedI simp: one_ereal_def emeasure_eq_measure) 
+
+lemma (in prob_space) erlang_distributed_mult_const:
+  assumes erlX: "distributed M lborel X (erlang_density k l)"
+  assumes a_pos[arith]: "0 < \<alpha>"  "0 < l"
+  shows  "distributed M lborel (\<lambda>x. \<alpha> * X x) (erlang_density k (l / \<alpha>))"
+proof (subst erlang_distributed_iff, safe)
+  have [measurable]: "random_variable borel X"  and  [arith]: "0 < l " 
+  and  [simp]: "\<And>a. 0 \<le> a \<Longrightarrow> prob {x \<in> space M. X x \<le> a} = erlang_CDF k l a"
+    by(insert erlX, auto simp: erlang_distributed_iff)
+
+  show "random_variable borel (\<lambda>x. \<alpha> * X x)" "0 < l / \<alpha>"  "0 < l / \<alpha>" 
+    by (auto simp:field_simps)
+  
+  fix a:: real assume [arith]: "0 \<le> a"
+  obtain b:: real  where [simp, arith]: "b = a/ \<alpha>" by blast 
+
+  have [arith]: "0 \<le> b" by (auto simp: divide_nonneg_pos)
+ 
+  have "prob {x \<in> space M. \<alpha> * X x \<le> a}  = prob {x \<in> space M.  X x \<le> b}"
+    by (rule arg_cong[where f= prob]) (auto simp:field_simps)
+  
+  moreover have "prob {x \<in> space M. X x \<le> b} = erlang_CDF k l b" by auto
+  moreover have "erlang_CDF k (l / \<alpha>) a = erlang_CDF k l b" unfolding erlang_CDF_def by auto
+  ultimately show "prob {x \<in> space M. \<alpha> * X x \<le> a} = erlang_CDF k (l / \<alpha>) a" by fastforce  
+qed
+
+lemma (in prob_space) has_bochner_integral_erlang_ith_moment:
+  fixes k i :: nat and l :: real
+  assumes [arith]: "0 < l" and D: "distributed M lborel X (erlang_density k l)"
+  shows "has_bochner_integral M (\<lambda>x. X x ^ i) (fact (k + i) / (fact k * l ^ i))"
+proof (rule has_bochner_integral_nn_integral)
+  show "AE x in M. 0 \<le> X x ^ i"
+    by (subst distributed_AE2[OF D]) (auto simp: erlang_density_def)
+  show "(\<integral>\<^sup>+ x. ereal (X x ^ i) \<partial>M) = ereal (fact (k + i) / (fact k * l ^ i))"
+    using nn_integral_erlang_ith_moment[of l k i]
+    by (subst distributed_nn_integral[symmetric, OF D]) auto
+qed (insert distributed_measurable[OF D], simp)
+
+lemma (in prob_space) erlang_ith_moment_integrable:
+  "0 < l \<Longrightarrow> distributed M lborel X (erlang_density k l) \<Longrightarrow> integrable M (\<lambda>x. X x ^ i)"
+  by rule (rule has_bochner_integral_erlang_ith_moment)
+
+lemma (in prob_space) erlang_ith_moment:
+  "0 < l \<Longrightarrow> distributed M lborel X (erlang_density k l) \<Longrightarrow>
+    expectation (\<lambda>x. X x ^ i) = fact (k + i) / (fact k * l ^ i)"
+  by (rule has_bochner_integral_integral_eq) (rule has_bochner_integral_erlang_ith_moment)
+
+lemma (in prob_space) erlang_distributed_variance:
+  assumes [arith]: "0 < l" and "distributed M lborel X (erlang_density k l)"
+  shows "variance X = (k + 1) / l\<^sup>2"
+proof (subst variance_eq)
+  show "integrable M X" "integrable M (\<lambda>x. (X x)\<^sup>2)"
+    using erlang_ith_moment_integrable[OF assms, of 1] erlang_ith_moment_integrable[OF assms, of 2]
+    by auto
+
+  show "expectation (\<lambda>x. (X x)\<^sup>2) - (expectation X)\<^sup>2 = real (k + 1) / l\<^sup>2"
+    using erlang_ith_moment[OF assms, of 1] erlang_ith_moment[OF assms, of 2]
+    by simp (auto simp: power2_eq_square field_simps real_of_nat_Suc)
+qed
+
 subsection {* Exponential distribution *}
 
-definition exponential_density :: "real \<Rightarrow> real \<Rightarrow> real" where
+abbreviation exponential_density :: "real \<Rightarrow> real \<Rightarrow> real" where
+  "exponential_density \<equiv> erlang_density 0"
+
+lemma exponential_density_def:
   "exponential_density l x = (if x < 0 then 0 else l * exp (- x * l))"
-
-lemma borel_measurable_exponential_density[measurable]: "exponential_density l \<in> borel_measurable borel"
-  by (auto simp add: exponential_density_def[abs_def])
+  by (simp add: fun_eq_iff erlang_density_def)
+
+lemma erlang_CDF_0: "erlang_CDF 0 l a = (if 0 \<le> a then 1 - exp (- l * a) else 0)"
+  by (simp add: erlang_CDF_def)
 
 lemma (in prob_space) exponential_distributed_params:
   assumes D: "distributed M lborel X (exponential_density l)"
   shows "0 < l"
 proof (cases l "0 :: real" rule: linorder_cases)

   from X.emeasure_space_1
   show "0 < l"
     by (simp add: emeasure_density distributed_distr_eq_density[OF D])
 qed assumption
 
-lemma
-  assumes [arith]: "0 < l"
-  shows emeasure_exponential_density_le0: "0 \<le> a \<Longrightarrow> emeasure (density lborel (exponential_density l)) {.. a} = 1 - exp (- a * l)"
-    and prob_space_exponential_density: "prob_space (density lborel (exponential_density l))"
-      (is "prob_space ?D")
-proof -
-  let ?f = "\<lambda>x. l * exp (- x * l)"
-  let ?F = "\<lambda>x. - exp (- x * l)"
-
-  have deriv: "\<And>x. DERIV ?F x :> ?f x" "\<And>x. 0 \<le> l * exp (- x * l)"
-    by (auto intro!: derivative_eq_intros simp: zero_le_mult_iff)
-
-  have "emeasure ?D (space ?D) = (\<integral>\<^sup>+ x. ereal (?f x) * indicator {0..} x \<partial>lborel)"
-    by (auto simp: emeasure_density exponential_density_def
-             intro!: nn_integral_cong split: split_indicator)
-  also have "\<dots> = ereal (0 - ?F 0)"
-  proof (rule nn_integral_FTC_atLeast)
-    have "((\<lambda>x. exp (l * x)) ---> 0) at_bot"
-      by (rule filterlim_compose[OF exp_at_bot filterlim_tendsto_pos_mult_at_bot[of _ l]])
-         (simp_all add: tendsto_const filterlim_ident)
-    then show "((\<lambda>x. - exp (- x * l)) ---> 0) at_top"
-      unfolding filterlim_at_top_mirror
-      by (simp add: tendsto_minus_cancel_left[symmetric] ac_simps)
-  qed (insert deriv, auto)
-  also have "\<dots> = 1" by (simp add: one_ereal_def)
-  finally have "emeasure ?D (space ?D) = 1" .
-  then show "prob_space ?D" by rule
-
-  assume "0 \<le> a"
-  have "emeasure ?D {..a} = (\<integral>\<^sup>+x. ereal (?f x) * indicator {0..a} x \<partial>lborel)"
-    by (auto simp add: emeasure_density intro!: nn_integral_cong split: split_indicator)
-       (auto simp: exponential_density_def)
-  also have "(\<integral>\<^sup>+x. ereal (?f x) * indicator {0..a} x \<partial>lborel) = ereal (?F a) - ereal (?F 0)"
-    using `0 \<le> a` deriv by (intro nn_integral_FTC_atLeastAtMost) auto
-  also have "\<dots> = 1 - exp (- a * l)"
-    by simp
-  finally show "emeasure ?D {.. a} = 1 - exp (- a * l)" .
-qed
-
+lemma prob_space_exponential_density: "0 < l \<Longrightarrow> prob_space (density lborel (exponential_density l))"
+  by (rule prob_space_erlang_density)
 
 lemma (in prob_space) exponential_distributedD_le:
   assumes D: "distributed M lborel X (exponential_density l)" and a: "0 \<le> a"
   shows "\<P>(x in M. X x \<le> a) = 1 - exp (- a * l)"
-proof -
-  have "emeasure M {x \<in> space M. X x \<le> a } = emeasure (distr M lborel X) {.. a}"
-    using distributed_measurable[OF D] 
-    by (subst emeasure_distr) (auto intro!: arg_cong2[where f=emeasure])
-  also have "\<dots> = emeasure (density lborel (exponential_density l)) {.. a}"
-    unfolding distributed_distr_eq_density[OF D] ..
-  also have "\<dots> = 1 - exp (- a * l)"
-    using emeasure_exponential_density_le0[OF exponential_distributed_params[OF D] a] .
-  finally show ?thesis
-    by (auto simp: measure_def)
-qed
+  using erlang_distributed_le[OF D exponential_distributed_params[OF D] a] a
+  by (simp add: erlang_CDF_def)
 
 lemma (in prob_space) exponential_distributedD_gt:
   assumes D: "distributed M lborel X (exponential_density l)" and a: "0 \<le> a"
   shows "\<P>(x in M. a < X x ) = exp (- a * l)"
-proof -
-  have "exp (- a * l) = 1 - \<P>(x in M. X x \<le> a)"
-    unfolding exponential_distributedD_le[OF D a] by simp
-  also have "\<dots> = prob (space M - {x \<in> space M. X x \<le> a })"
-    using distributed_measurable[OF D]
-    by (subst prob_compl) auto
-  also have "\<dots> = \<P>(x in M. a < X x )"
-    by (auto intro!: arg_cong[where f=prob] simp: not_le)
-  finally show ?thesis by simp
-qed
+  using erlang_distributed_gt[OF D exponential_distributed_params[OF D] a] a
+  by (simp add: erlang_CDF_def)
 
 lemma (in prob_space) exponential_distributed_memoryless:
   assumes D: "distributed M lborel X (exponential_density l)" and a: "0 \<le> a"and t: "0 \<le> t"
   shows "\<P>(x in M. a + t < X x \<bar> a < X x) = \<P>(x in M. t < X x)"
 proof -

 
 lemma exponential_distributedI:
   assumes X[measurable]: "X \<in> borel_measurable M" and [arith]: "0 < l"
     and X_distr: "\<And>a. 0 \<le> a \<Longrightarrow> emeasure M {x\<in>space M. X x \<le> a} = 1 - exp (- a * l)"
   shows "distributed M lborel X (exponential_density l)"
-proof (rule distributedI_borel_atMost)
-  fix a :: real
-
-  { assume "a \<le> 0"  
-    with X have "emeasure M {x\<in>space M. X x \<le> a} \<le> emeasure M {x\<in>space M. X x \<le> 0}"
-      by (intro emeasure_mono) auto
-    then have "emeasure M {x\<in>space M. X x \<le> a} = 0"
-      using X_distr[of 0] by (simp add: one_ereal_def emeasure_le_0_iff) }
-  note eq_0 = this
-
-  have "\<And>x. \<not> 0 \<le> a \<Longrightarrow> ereal (exponential_density l x) * indicator {..a} x = 0"
-    by (simp add: exponential_density_def)
-  then show "(\<integral>\<^sup>+ x. exponential_density l x * indicator {..a} x \<partial>lborel) = ereal (if 0 \<le> a then 1 - exp (- a * l) else 0)"
-    using emeasure_exponential_density_le0[of l a]
-    by (auto simp: emeasure_density times_ereal.simps[symmetric] ereal_indicator
-             simp del: times_ereal.simps ereal_zero_times)
-  show "emeasure M {x\<in>space M. X x \<le> a} = ereal (if 0 \<le> a then 1 - exp (- a * l) else 0)"
-    using X_distr[of a] eq_0 by (auto simp: one_ereal_def)
-  show "AE x in lborel. 0 \<le> exponential_density l x "
-    by (auto simp: exponential_density_def)
-qed simp_all
+proof (rule erlang_distributedI)
+  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)"
+    using X_distr[of a] by (simp add: erlang_CDF_def one_ereal_def)
+qed fact+
 
 lemma (in prob_space) exponential_distributed_iff:
   "distributed M lborel X (exponential_density l) \<longleftrightarrow>
     (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)))"
-  using
-    distributed_measurable[of M lborel X "exponential_density l"]
-    exponential_distributed_params[of X l]
-    emeasure_exponential_density_le0[of l]
-    exponential_distributedD_le[of X l]
-  by (auto intro!: exponential_distributedI simp: one_ereal_def emeasure_eq_measure)
-
-lemma borel_integral_x_exp:
-  "has_bochner_integral lborel (\<lambda>x. x * exp (- x) * indicator {0::real ..} x) 1"
-proof (rule has_bochner_integral_monotone_convergence)
-  let ?f = "\<lambda>i x. x * exp (- x) * indicator {0::real .. i} x"
-  have "eventually (\<lambda>b::real. 0 \<le> b) at_top"
-    by (rule eventually_ge_at_top)
-  then have "eventually (\<lambda>b. 1 - (inverse (exp b) + b / exp b) = integral\<^sup>L lborel (?f b)) at_top"
-  proof eventually_elim
-   fix b :: real assume [simp]: "0 \<le> b"
-    have "(\<integral>x. (exp (-x)) * indicator {0 .. b} x \<partial>lborel) - (integral\<^sup>L lborel (?f b)) = 
-      (\<integral>x. (exp (-x) - x * exp (-x)) * indicator {0 .. b} x \<partial>lborel)"
-      by (subst integral_diff[symmetric])
-         (auto intro!: borel_integrable_atLeastAtMost integral_cong split: split_indicator)
-    also have "\<dots> = b * exp (-b) - 0 * exp (- 0)"
-    proof (rule integral_FTC_atLeastAtMost)
-      fix x assume "0 \<le> x" "x \<le> b"
-      show "DERIV (\<lambda>x. x * exp (-x)) x :> exp (-x) - x * exp (-x)"
-        by (auto intro!: derivative_eq_intros)
-      show "isCont (\<lambda>x. exp (- x) - x * exp (- x)) x "
-        by (intro continuous_intros)
-    qed fact
-    also have "(\<integral>x. (exp (-x)) * indicator {0 .. b} x \<partial>lborel) = - exp (- b) - - exp (- 0)"
-      by (rule integral_FTC_atLeastAtMost) (auto intro!: derivative_eq_intros)
-    finally show "1 - (inverse (exp b) + b / exp b) = integral\<^sup>L lborel (?f b)"
-      by (auto simp: field_simps exp_minus inverse_eq_divide)
+  using exponential_distributed_params[of X l] erlang_distributed_iff[of l X 0] by (auto simp: erlang_CDF_0)
+
+
+lemma (in prob_space) exponential_distributed_expectation:
+  "distributed M lborel X (exponential_density l) \<Longrightarrow> expectation X = 1 / l"
+  using erlang_ith_moment[OF exponential_distributed_params, of X l X 0 1] by simp
+
+lemma exponential_density_nonneg: "0 < l \<Longrightarrow> 0 \<le> exponential_density l x"
+  by (auto simp: exponential_density_def)
+
+lemma (in prob_space) exponential_distributed_min:
+  assumes expX: "distributed M lborel X (exponential_density l)"
+  assumes expY: "distributed M lborel Y (exponential_density u)"
+  assumes ind: "indep_var borel X borel Y"
+  shows "distributed M lborel (\<lambda>x. min (X x) (Y x)) (exponential_density (l + u))"
+proof (subst exponential_distributed_iff, safe)
+  have randX: "random_variable borel X" using expX by (simp add: exponential_distributed_iff)
+  moreover have randY: "random_variable borel Y" using expY by (simp add: exponential_distributed_iff)
+  ultimately show "random_variable borel (\<lambda>x. min (X x) (Y x))" by auto
+  
+  have "0 < l" by (rule exponential_distributed_params) fact
+  moreover have "0 < u" by (rule exponential_distributed_params) fact
+  ultimately  show " 0 < l + u" by force
+
+  fix a::real assume a[arith]: "0 \<le> a"
+  have gt1[simp]: "\<P>(x in M. a < X x ) = exp (- a * l)" by (rule exponential_distributedD_gt[OF expX a]) 
+  have gt2[simp]: "\<P>(x in M. a < Y x ) = exp (- a * u)" by (rule exponential_distributedD_gt[OF expY a]) 
+
+  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])
+
+  also have " ... =  \<P>(x in M. a < (X x)) *  \<P>(x in M. a< (Y x) )"
+    using prob_indep_random_variable[OF ind, of "{a <..}" "{a <..}"] by simp
+  also have " ... = exp (- a * (l + u))" by (auto simp:field_simps mult_exp_exp)
+  finally have indep_prob: "\<P>(x in M. a < (min (X x) (Y x)) ) = exp (- a * (l + u))" .
+
+  have "{x \<in> space M. (min (X x) (Y x)) \<le>a } = (space M - {x \<in> space M. a<(min (X x) (Y x)) })"
+    by auto
+  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}"
+    using randX randY by (auto simp: prob_compl) 
+  then show "prob {x \<in> space M. (min (X x) (Y x)) \<le> a} = 1 - exp (- a * (l + u))"
+    using indep_prob by auto
+qed
+ 
+lemma (in prob_space) exponential_distributed_Min:
+  assumes finI: "finite I"
+  assumes A: "I \<noteq> {}"
+  assumes expX: "\<And>i. i \<in> I \<Longrightarrow> distributed M lborel (X i) (exponential_density (l i))"
+  assumes ind: "indep_vars (\<lambda>i. borel) X I" 
+  shows "distributed M lborel (\<lambda>x. Min ((\<lambda>i. X i x)`I)) (exponential_density (\<Sum>i\<in>I. l i))"
+using assms
+proof (induct rule: finite_ne_induct)
+  case (singleton i) then show ?case by simp
+next
+  case (insert i I)
+  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)))"
+      by (intro exponential_distributed_min indep_vars_Min insert)
+         (auto intro: indep_vars_subset) 
+  then show ?case
+    using insert by simp
+qed
+
+lemma (in prob_space) exponential_distributed_variance:
+  "distributed M lborel X (exponential_density l) \<Longrightarrow> variance X = 1 / l\<^sup>2"
+  using erlang_distributed_variance[OF exponential_distributed_params, of X l X 0] by simp
+
+lemma nn_integral_zero': "AE x in M. f x = 0 \<Longrightarrow> (\<integral>\<^sup>+x. f x \<partial>M) = 0"
+  by (simp cong: nn_integral_cong_AE)
+
+lemma convolution_erlang_density:
+  fixes k\<^sub>1 k\<^sub>2 :: nat
+  assumes [simp, arith]: "0 < l"
+  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) =
+    (erlang_density (Suc k\<^sub>1 + Suc k\<^sub>2 - 1) l)"
+      (is "?LHS = ?RHS")
+proof
+  fix x :: real
+  have "x \<le> 0 \<or> 0 < x"
+    by arith
+  then show "?LHS x = ?RHS x"
+  proof
+    assume "x \<le> 0" then show ?thesis
+      apply (subst nn_integral_zero')
+      apply (rule AE_I[where N="{0}"])
+      apply (auto simp add: erlang_density_def not_less)
+      done
+  next
+    note zero_le_mult_iff[simp] zero_le_divide_iff[simp]
+  
+    have I_eq1: "integral\<^sup>N lborel (erlang_density (Suc k\<^sub>1 + Suc k\<^sub>2 - 1) l) = 1"
+      using nn_integral_erlang_ith_moment[of l "Suc k\<^sub>1 + Suc k\<^sub>2 - 1" 0] by (simp del: fact_Suc)
+  
+    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"
+      apply (subst I_eq1[symmetric])
+      unfolding erlang_density_def
+      by (auto intro!: nn_integral_cong split:split_indicator)
+  
+    have "prob_space (density lborel ?LHS)"
+      unfolding times_ereal.simps[symmetric]
+      by (intro prob_space_convolution_density) 
+         (auto intro!: prob_space_erlang_density erlang_density_nonneg)
+    then have 2: "integral\<^sup>N lborel ?LHS = 1"
+      by (auto dest!: prob_space.emeasure_space_1 simp: emeasure_density)
+  
+    let ?I = "(integral\<^sup>N lborel (\<lambda>y. ereal ((1 - y)^ k\<^sub>1 * y^k\<^sub>2 * indicator {0..1} y)))"
+    let ?C = "real (fact (Suc (k\<^sub>1 + k\<^sub>2))) / (real (fact k\<^sub>1) * real (fact k\<^sub>2))"
+    let ?s = "Suc k\<^sub>1 + Suc k\<^sub>2 - 1"
+    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)"
+
+    { fix x :: real assume [arith]: "0 < x"
+      have *: "\<And>x y n. (x - y * x::real)^n = x^n * (1 - y)^n"
+        unfolding power_mult_distrib[symmetric] by (simp add: field_simps)
+    
+      have "?LHS x = ?L x"
+        unfolding erlang_density_def
+        by (auto intro!: nn_integral_cong split:split_indicator)
+      also have "... = (\<lambda>x. ereal ?C * ?I * erlang_density ?s l x) x"
+        apply (subst nn_integral_real_affine[where c=x and t = 0])
+        apply (simp_all add: nn_integral_cmult[symmetric] nn_integral_multc[symmetric] erlang_density_nonneg del: fact_Suc)
+        apply (intro nn_integral_cong)
+        apply (auto simp add: erlang_density_def mult_less_0_iff exp_minus field_simps exp_diff power_add *
+                    simp del: fact_Suc split: split_indicator)
+        done
+      finally have "(\<integral>\<^sup>+y. ereal (erlang_density k\<^sub>1 l (x - y) * erlang_density k\<^sub>2 l y) \<partial>lborel) = 
+        (\<lambda>x. ereal ?C * ?I * erlang_density ?s l x) x"
+        by simp }
+    note * = this
+
+    assume [arith]: "0 < x"
+    have 3: "1 = integral\<^sup>N lborel (\<lambda>xa. ?LHS xa * indicator {0<..} xa)"
+      by (subst 2[symmetric])
+         (auto intro!: nn_integral_cong_AE AE_I[where N="{0}"]
+               simp: erlang_density_def  nn_integral_multc[symmetric] indicator_def split: split_if_asm)
+    also have "... = integral\<^sup>N lborel (\<lambda>x. (ereal (?C) * ?I) * ((erlang_density ?s l x) * indicator {0<..} x))"
+      by (auto intro!: nn_integral_cong simp: * split: split_indicator)
+    also have "... = ereal (?C) * ?I"
+      using 1
+      by (auto simp: nn_integral_nonneg nn_integral_cmult)  
+    finally have " ereal (?C) * ?I = 1" by presburger
+  
+    then show ?thesis
+      using * by simp
   qed
-  then have "((\<lambda>i. integral\<^sup>L lborel (?f i)) ---> 1 - (0 + 0)) at_top"
-  proof (rule Lim_transform_eventually)
-    show "((\<lambda>i. 1 - (inverse (exp i) + i / exp i)) ---> 1 - (0 + 0 :: real)) at_top"
-      using tendsto_power_div_exp_0[of 1]
-      by (intro tendsto_intros tendsto_inverse_0_at_top exp_at_top) simp
-  qed
-  then show "(\<lambda>i. integral\<^sup>L lborel (?f i)) ----> 1"
-    by (intro filterlim_compose[OF _ filterlim_real_sequentially]) simp
-  show "AE x in lborel. mono (\<lambda>n::nat. x * exp (- x) * indicator {0..real n} x)"
-    by (auto simp: mono_def mult_le_0_iff zero_le_mult_iff split: split_indicator)
-  show "\<And>i::nat. integrable lborel (\<lambda>x. x * exp (- x) * indicator {0..real i} x)"
-    by (rule borel_integrable_atLeastAtMost) auto
-  show "AE x in lborel. (\<lambda>i. x * exp (- x) * indicator {0..real i} x) ----> x * exp (- x) * indicator {0..} x"
-    apply (intro AE_I2 Lim_eventually )
-    apply (rule_tac c="natfloor x + 1" in eventually_sequentiallyI)
-    apply (auto simp add: not_le dest!: ge_natfloor_plus_one_imp_gt[simplified] split: split_indicator)
-    done
-qed (auto intro!: borel_measurable_times borel_measurable_exp)
-
-lemma (in prob_space) exponential_distributed_expectation:
-  assumes D: "distributed M lborel X (exponential_density l)"
-  shows "expectation X = 1 / l"
-proof (subst distributed_integral[OF D, of "\<lambda>x. x", symmetric])
-  have "0 < l"
-   using exponential_distributed_params[OF D] .
-  have [simp]: "\<And>x. x * (l * (exp (- (x * l)) * indicator {0..} (x * l))) =
-    x * exponential_density l x"
-    using `0 < l`
-    by (auto split: split_indicator simp: zero_le_mult_iff exponential_density_def)
-  from borel_integral_x_exp `0 < l`
-  have "has_bochner_integral lborel (\<lambda>x. exponential_density l x * x) (1 / l)"
-    by (subst (asm) lborel_has_bochner_integral_real_affine_iff[of l _ _ 0])
-       (simp_all add: field_simps)
-  then show "(\<integral> x. exponential_density l x * x \<partial>lborel) = 1 / l"
-    by (metis has_bochner_integral_integral_eq)
-qed simp
+qed
+
+lemma (in prob_space) sum_indep_erlang:
+  assumes indep: "indep_var borel X borel Y"
+  assumes [simp, arith]: "0 < l"
+  assumes erlX: "distributed M lborel X (erlang_density k\<^sub>1 l)"
+  assumes erlY: "distributed M lborel Y (erlang_density k\<^sub>2 l)"
+  shows "distributed M lborel (\<lambda>x. X x + Y x) (erlang_density (Suc k\<^sub>1 + Suc k\<^sub>2 - 1) l)"
+  using assms
+  apply (subst convolution_erlang_density[symmetric, OF `0<l`])
+  apply (intro distributed_convolution)
+  apply auto
+  done
+
+lemma (in prob_space) erlang_distributed_setsum:
+  assumes finI : "finite I"
+  assumes A: "I \<noteq> {}"
+  assumes [simp, arith]: "0 < l"
+  assumes expX: "\<And>i. i \<in> I \<Longrightarrow> distributed M lborel (X i) (erlang_density (k i) l)"
+  assumes ind: "indep_vars (\<lambda>i. borel) X I"
+  shows "distributed M lborel (\<lambda>x. \<Sum>i\<in>I. X i x) (erlang_density ((\<Sum>i\<in>I. Suc (k i)) - 1) l)"
+using assms
+proof (induct rule: finite_ne_induct)
+  case (singleton i) then show ?case by auto
+next
+  case (insert i I)
+    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)"
+      by(intro sum_indep_erlang indep_vars_setsum) (auto intro!: indep_vars_subset)
+    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)"
+      using insert by auto
+    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"
+      using insert by (auto intro!: Suc_pred simp: ac_simps)    
+    finally show ?case by fast
+qed
+
+lemma (in prob_space) exponential_distributed_setsum:
+  assumes finI: "finite I"
+  assumes A: "I \<noteq> {}"
+  assumes expX: "\<And>i. i \<in> I \<Longrightarrow> distributed M lborel (X i) (exponential_density l)"
+  assumes ind: "indep_vars (\<lambda>i. borel) X I" 
+  shows "distributed M lborel (\<lambda>x. \<Sum>i\<in>I. X i x) (erlang_density ((card I) - 1) l)"
+proof -
+  obtain i where "i \<in> I" using assms by auto
+  note exponential_distributed_params[OF expX[OF this]]
+  from erlang_distributed_setsum[OF assms(1,2) this assms(3,4)] show ?thesis by simp
+qed
 
 subsection {* Uniform distribution *}
 
 lemma uniform_distrI:
   assumes X: "X \<in> measurable M M'"

       unfolding * square_diff_square_factored by (auto simp: field_simps)
   qed (insert `a < b`, simp)
   finally show "(\<integral> x. indicator {a .. b} x / measure lborel {a .. b} * x \<partial>lborel) = (a + b) / 2" .
 qed auto
 
+lemma (in prob_space) uniform_distributed_variance:
+  fixes a b :: real
+  assumes D: "distributed M lborel X (\<lambda>x. indicator {a .. b} x / measure lborel {a .. b})"
+  shows "variance X = (b - a)\<^sup>2 / 12"
+proof (subst distributed_variance)
+  have [arith]: "a < b" using uniform_distributed_bounds[OF D] .
+  let ?\<mu> = "expectation X" let ?D = "\<lambda>x. indicator {a..b} (x + ?\<mu>) / measure lborel {a..b}"
+  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)"
+    by (intro integral_cong) (auto split: split_indicator)
+  also have "\<dots> = (b - a)\<^sup>2 / 12"
+    by (simp add: integral_power measure_lebesgue_Icc uniform_distributed_expectation[OF D])
+       (simp add: eval_nat_numeral field_simps )
+  finally show "(\<integral>x. x\<^sup>2 * ?D x \<partial>lborel) = (b - a)\<^sup>2 / 12" .
+qed fact
+
 end