src/HOL/Probability/Distributions.thy
author hoelzl
Mon Apr 14 13:08:17 2014 +0200 (2014-04-14)
changeset 56571 f4635657d66f
parent 56536 aefb4a8da31f
child 56993 e5366291d6aa
permissions -rw-r--r--
added divide_nonneg_nonneg and co; made it a simp rule
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theory Distributions
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  imports Probability_Measure
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begin
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subsection {* Exponential distribution *}
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definition exponential_density :: "real \<Rightarrow> real \<Rightarrow> real" where
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  "exponential_density l x = (if x < 0 then 0 else l * exp (- x * l))"
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lemma borel_measurable_exponential_density[measurable]: "exponential_density l \<in> borel_measurable borel"
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  by (auto simp add: exponential_density_def[abs_def])
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lemma (in prob_space) exponential_distributed_params:
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  assumes D: "distributed M lborel X (exponential_density l)"
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  shows "0 < l"
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proof (cases l "0 :: real" rule: linorder_cases)
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  assume "l < 0"
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  have "emeasure lborel {0 <.. 1::real} \<le>
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    emeasure lborel {x :: real \<in> space lborel. 0 < x}"
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    by (rule emeasure_mono) (auto simp: greaterThan_def[symmetric])
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  also have "emeasure lborel {x :: real \<in> space lborel. 0 < x} = 0"
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  proof -
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    have "AE x in lborel. 0 \<le> exponential_density l x"
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      using assms by (auto simp: distributed_real_AE)
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    then have "AE x in lborel. x \<le> (0::real)"
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      apply eventually_elim 
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      using `l < 0`
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      apply (auto simp: exponential_density_def zero_le_mult_iff split: split_if_asm)
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      done
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    then show "emeasure lborel {x :: real \<in> space lborel. 0 < x} = 0"
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      by (subst (asm) AE_iff_measurable[OF _ refl]) (auto simp: not_le greaterThan_def[symmetric])
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  qed
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  finally show "0 < l" by simp
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next
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  assume "l = 0"
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  then have [simp]: "\<And>x. ereal (exponential_density l x) = 0"
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    by (simp add: exponential_density_def)
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  interpret X: prob_space "distr M lborel X"
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    using distributed_measurable[OF D] by (rule prob_space_distr)
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  from X.emeasure_space_1
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  show "0 < l"
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    by (simp add: emeasure_density distributed_distr_eq_density[OF D])
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qed assumption
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lemma
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  assumes [arith]: "0 < l"
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  shows emeasure_exponential_density_le0: "0 \<le> a \<Longrightarrow> emeasure (density lborel (exponential_density l)) {.. a} = 1 - exp (- a * l)"
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    and prob_space_exponential_density: "prob_space (density lborel (exponential_density l))"
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      (is "prob_space ?D")
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proof -
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  let ?f = "\<lambda>x. l * exp (- x * l)"
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  let ?F = "\<lambda>x. - exp (- x * l)"
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  have deriv: "\<And>x. DERIV ?F x :> ?f x" "\<And>x. 0 \<le> l * exp (- x * l)"
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    by (auto intro!: derivative_eq_intros simp: zero_le_mult_iff)
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  have "emeasure ?D (space ?D) = (\<integral>\<^sup>+ x. ereal (?f x) * indicator {0..} x \<partial>lborel)"
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    by (auto simp: emeasure_density exponential_density_def
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             intro!: positive_integral_cong split: split_indicator)
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  also have "\<dots> = ereal (0 - ?F 0)"
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  proof (rule positive_integral_FTC_atLeast)
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    have "((\<lambda>x. exp (l * x)) ---> 0) at_bot"
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      by (rule filterlim_compose[OF exp_at_bot filterlim_tendsto_pos_mult_at_bot[of _ l]])
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         (simp_all add: tendsto_const filterlim_ident)
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    then show "((\<lambda>x. - exp (- x * l)) ---> 0) at_top"
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      unfolding filterlim_at_top_mirror
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      by (simp add: tendsto_minus_cancel_left[symmetric] ac_simps)
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  qed (insert deriv, auto)
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  also have "\<dots> = 1" by (simp add: one_ereal_def)
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  finally have "emeasure ?D (space ?D) = 1" .
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  then show "prob_space ?D" by rule
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  assume "0 \<le> a"
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  have "emeasure ?D {..a} = (\<integral>\<^sup>+x. ereal (?f x) * indicator {0..a} x \<partial>lborel)"
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    by (auto simp add: emeasure_density intro!: positive_integral_cong split: split_indicator)
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       (auto simp: exponential_density_def)
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  also have "(\<integral>\<^sup>+x. ereal (?f x) * indicator {0..a} x \<partial>lborel) = ereal (?F a) - ereal (?F 0)"
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    using `0 \<le> a` deriv by (intro positive_integral_FTC_atLeastAtMost) auto
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  also have "\<dots> = 1 - exp (- a * l)"
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    by simp
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  finally show "emeasure ?D {.. a} = 1 - exp (- a * l)" .
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qed
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lemma (in prob_space) exponential_distributedD_le:
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  assumes D: "distributed M lborel X (exponential_density l)" and a: "0 \<le> a"
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  shows "\<P>(x in M. X x \<le> a) = 1 - exp (- a * l)"
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proof -
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  have "emeasure M {x \<in> space M. X x \<le> a } = emeasure (distr M lborel X) {.. a}"
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    using distributed_measurable[OF D] 
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    by (subst emeasure_distr) (auto intro!: arg_cong2[where f=emeasure])
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  also have "\<dots> = emeasure (density lborel (exponential_density l)) {.. a}"
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    unfolding distributed_distr_eq_density[OF D] ..
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  also have "\<dots> = 1 - exp (- a * l)"
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    using emeasure_exponential_density_le0[OF exponential_distributed_params[OF D] a] .
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  finally show ?thesis
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    by (auto simp: measure_def)
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qed
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lemma (in prob_space) exponential_distributedD_gt:
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  assumes D: "distributed M lborel X (exponential_density l)" and a: "0 \<le> a"
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  shows "\<P>(x in M. a < X x ) = exp (- a * l)"
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proof -
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  have "exp (- a * l) = 1 - \<P>(x in M. X x \<le> a)"
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    unfolding exponential_distributedD_le[OF D a] by simp
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  also have "\<dots> = prob (space M - {x \<in> space M. X x \<le> a })"
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    using distributed_measurable[OF D]
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    by (subst prob_compl) auto
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  also have "\<dots> = \<P>(x in M. a < X x )"
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    by (auto intro!: arg_cong[where f=prob] simp: not_le)
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  finally show ?thesis by simp
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qed
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lemma (in prob_space) exponential_distributed_memoryless:
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  assumes D: "distributed M lborel X (exponential_density l)" and a: "0 \<le> a"and t: "0 \<le> t"
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  shows "\<P>(x in M. a + t < X x \<bar> a < X x) = \<P>(x in M. t < X x)"
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proof -
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  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)"
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    using `0 \<le> t` by (auto simp: cond_prob_def intro!: arg_cong[where f=prob] arg_cong2[where f="op /"])
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  also have "\<dots> = exp (- (a + t) * l) / exp (- a * l)"
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    using a t by (simp add: exponential_distributedD_gt[OF D])
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  also have "\<dots> = exp (- t * l)"
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    using exponential_distributed_params[OF D] by (auto simp: field_simps exp_add[symmetric])
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  finally show ?thesis
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    using t by (simp add: exponential_distributedD_gt[OF D])
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qed
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lemma exponential_distributedI:
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  assumes X[measurable]: "X \<in> borel_measurable M" and [arith]: "0 < l"
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    and X_distr: "\<And>a. 0 \<le> a \<Longrightarrow> emeasure M {x\<in>space M. X x \<le> a} = 1 - exp (- a * l)"
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  shows "distributed M lborel X (exponential_density l)"
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proof (rule distributedI_borel_atMost)
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  fix a :: real
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  { assume "a \<le> 0"  
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    with X have "emeasure M {x\<in>space M. X x \<le> a} \<le> emeasure M {x\<in>space M. X x \<le> 0}"
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      by (intro emeasure_mono) auto
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    then have "emeasure M {x\<in>space M. X x \<le> a} = 0"
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      using X_distr[of 0] by (simp add: one_ereal_def emeasure_le_0_iff) }
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  note eq_0 = this
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  have "\<And>x. \<not> 0 \<le> a \<Longrightarrow> ereal (exponential_density l x) * indicator {..a} x = 0"
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    by (simp add: exponential_density_def)
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  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)"
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    using emeasure_exponential_density_le0[of l a]
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    by (auto simp: emeasure_density times_ereal.simps[symmetric] ereal_indicator
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             simp del: times_ereal.simps ereal_zero_times)
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  show "emeasure M {x\<in>space M. X x \<le> a} = ereal (if 0 \<le> a then 1 - exp (- a * l) else 0)"
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    using X_distr[of a] eq_0 by (auto simp: one_ereal_def)
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  show "AE x in lborel. 0 \<le> exponential_density l x "
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    by (auto simp: exponential_density_def)
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qed simp_all
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lemma (in prob_space) exponential_distributed_iff:
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  "distributed M lborel X (exponential_density l) \<longleftrightarrow>
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    (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)))"
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  using
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    distributed_measurable[of M lborel X "exponential_density l"]
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    exponential_distributed_params[of X l]
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    emeasure_exponential_density_le0[of l]
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    exponential_distributedD_le[of X l]
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  by (auto intro!: exponential_distributedI simp: one_ereal_def emeasure_eq_measure)
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lemma borel_integral_x_exp:
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  "(\<integral>x. x * exp (- x) * indicator {0::real ..} x \<partial>lborel) = 1"
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proof (rule integral_monotone_convergence)
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  let ?f = "\<lambda>i x. x * exp (- x) * indicator {0::real .. i} x"
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  have "eventually (\<lambda>b::real. 0 \<le> b) at_top"
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    by (rule eventually_ge_at_top)
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  then have "eventually (\<lambda>b. 1 - (inverse (exp b) + b / exp b) = integral\<^sup>L lborel (?f b)) at_top"
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  proof eventually_elim
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   fix b :: real assume [simp]: "0 \<le> b"
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    have "(\<integral>x. (exp (-x)) * indicator {0 .. b} x \<partial>lborel) - (integral\<^sup>L lborel (?f b)) = 
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      (\<integral>x. (exp (-x) - x * exp (-x)) * indicator {0 .. b} x \<partial>lborel)"
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      by (subst integral_diff(2)[symmetric])
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         (auto intro!: borel_integrable_atLeastAtMost integral_cong split: split_indicator)
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    also have "\<dots> = b * exp (-b) - 0 * exp (- 0)"
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    proof (rule integral_FTC_atLeastAtMost)
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      fix x assume "0 \<le> x" "x \<le> b"
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      show "DERIV (\<lambda>x. x * exp (-x)) x :> exp (-x) - x * exp (-x)"
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        by (auto intro!: derivative_eq_intros)
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      show "isCont (\<lambda>x. exp (- x) - x * exp (- x)) x "
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        by (intro continuous_intros)
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    qed fact
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    also have "(\<integral>x. (exp (-x)) * indicator {0 .. b} x \<partial>lborel) = - exp (- b) - - exp (- 0)"
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      by (rule integral_FTC_atLeastAtMost) (auto intro!: derivative_eq_intros)
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    finally show "1 - (inverse (exp b) + b / exp b) = integral\<^sup>L lborel (?f b)"
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      by (auto simp: field_simps exp_minus inverse_eq_divide)
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  qed
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  then have "((\<lambda>i. integral\<^sup>L lborel (?f i)) ---> 1 - (0 + 0)) at_top"
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  proof (rule Lim_transform_eventually)
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    show "((\<lambda>i. 1 - (inverse (exp i) + i / exp i)) ---> 1 - (0 + 0 :: real)) at_top"
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      using tendsto_power_div_exp_0[of 1]
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      by (intro tendsto_intros tendsto_inverse_0_at_top exp_at_top) simp
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  qed
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  then show "(\<lambda>i. integral\<^sup>L lborel (?f i)) ----> 1"
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    by (intro filterlim_compose[OF _ filterlim_real_sequentially]) simp
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  show "AE x in lborel. mono (\<lambda>n::nat. x * exp (- x) * indicator {0..real n} x)"
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    by (auto simp: mono_def mult_le_0_iff zero_le_mult_iff split: split_indicator)
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  show "\<And>i::nat. integrable lborel (\<lambda>x. x * exp (- x) * indicator {0..real i} x)"
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    by (rule borel_integrable_atLeastAtMost) auto
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  show "AE x in lborel. (\<lambda>i. x * exp (- x) * indicator {0..real i} x) ----> x * exp (- x) * indicator {0..} x"
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    apply (intro AE_I2 Lim_eventually )
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    apply (rule_tac c="natfloor x + 1" in eventually_sequentiallyI)
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    apply (auto simp add: not_le dest!: ge_natfloor_plus_one_imp_gt[simplified] split: split_indicator)
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    done
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qed (auto intro!: borel_measurable_times borel_measurable_exp)
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lemma (in prob_space) exponential_distributed_expectation:
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  assumes D: "distributed M lborel X (exponential_density l)"
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  shows "expectation X = 1 / l"
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proof (subst distributed_integral[OF D, of "\<lambda>x. x", symmetric])
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  have "0 < l"
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   using exponential_distributed_params[OF D] .
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  have [simp]: "\<And>x. x * (l * (exp (- (x * l)) * indicator {0..} (x * l))) =
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    x * exponential_density l x"
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    using `0 < l`
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    by (auto split: split_indicator simp: zero_le_mult_iff exponential_density_def)
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  from borel_integral_x_exp `0 < l`
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  show "(\<integral> x. exponential_density l x * x \<partial>lborel) = 1 / l"
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    by (subst (asm) lebesgue_integral_real_affine[of "l" _ 0])
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       (simp_all add: borel_measurable_exp nonzero_eq_divide_eq ac_simps)
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qed simp
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subsection {* Uniform distribution *}
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lemma uniform_distrI:
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  assumes X: "X \<in> measurable M M'"
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    and A: "A \<in> sets M'" "emeasure M' A \<noteq> \<infinity>" "emeasure M' A \<noteq> 0"
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  assumes distr: "\<And>B. B \<in> sets M' \<Longrightarrow> emeasure M (X -` B \<inter> space M) = emeasure M' (A \<inter> B) / emeasure M' A"
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  shows "distr M M' X = uniform_measure M' A"
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  unfolding uniform_measure_def
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proof (intro measure_eqI)
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  let ?f = "\<lambda>x. indicator A x / emeasure M' A"
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  fix B assume B: "B \<in> sets (distr M M' X)"
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  with X have "emeasure M (X -` B \<inter> space M) = emeasure M' (A \<inter> B) / emeasure M' A"
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    by (simp add: distr[of B] measurable_sets)
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  also have "\<dots> = (1 / emeasure M' A) * emeasure M' (A \<inter> B)"
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     by simp
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  also have "\<dots> = (\<integral>\<^sup>+ x. (1 / emeasure M' A) * indicator (A \<inter> B) x \<partial>M')"
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    using A B
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    by (intro positive_integral_cmult_indicator[symmetric]) (auto intro!: zero_le_divide_ereal)
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  also have "\<dots> = (\<integral>\<^sup>+ x. ?f x * indicator B x \<partial>M')"
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    by (rule positive_integral_cong) (auto split: split_indicator)
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  finally show "emeasure (distr M M' X) B = emeasure (density M' ?f) B"
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   246
    using A B X by (auto simp add: emeasure_distr emeasure_density)
hoelzl@50419
   247
qed simp
hoelzl@50419
   248
hoelzl@50419
   249
lemma uniform_distrI_borel:
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   250
  fixes A :: "real set"
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   251
  assumes X[measurable]: "X \<in> borel_measurable M" and A: "emeasure lborel A = ereal r" "0 < r"
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   252
    and [measurable]: "A \<in> sets borel"
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   253
  assumes distr: "\<And>a. emeasure M {x\<in>space M. X x \<le> a} = emeasure lborel (A \<inter> {.. a}) / r"
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   254
  shows "distributed M lborel X (\<lambda>x. indicator A x / measure lborel A)"
hoelzl@50419
   255
proof (rule distributedI_borel_atMost)
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   256
  let ?f = "\<lambda>x. 1 / r * indicator A x"
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   257
  fix a
hoelzl@50419
   258
  have "emeasure lborel (A \<inter> {..a}) \<le> emeasure lborel A"
hoelzl@50419
   259
    using A by (intro emeasure_mono) auto
hoelzl@50419
   260
  also have "\<dots> < \<infinity>"
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   261
    using A by simp
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   262
  finally have fin: "emeasure lborel (A \<inter> {..a}) \<noteq> \<infinity>"
hoelzl@50419
   263
    by simp
hoelzl@50419
   264
  from emeasure_eq_ereal_measure[OF this]
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   265
  have fin_eq: "emeasure lborel (A \<inter> {..a}) / r = ereal (measure lborel (A \<inter> {..a}) / r)"
hoelzl@50419
   266
    using A by simp
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   267
  then show "emeasure M {x\<in>space M. X x \<le> a} = ereal (measure lborel (A \<inter> {..a}) / r)"
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   268
    using distr by simp
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   269
 
wenzelm@53015
   270
  have "(\<integral>\<^sup>+ x. ereal (indicator A x / measure lborel A * indicator {..a} x) \<partial>lborel) =
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   271
    (\<integral>\<^sup>+ x. ereal (1 / measure lborel A) * indicator (A \<inter> {..a}) x \<partial>lborel)"
hoelzl@50419
   272
    by (auto intro!: positive_integral_cong split: split_indicator)
hoelzl@50419
   273
  also have "\<dots> = ereal (1 / measure lborel A) * emeasure lborel (A \<inter> {..a})"
hoelzl@50419
   274
    using `A \<in> sets borel`
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   275
    by (intro positive_integral_cmult_indicator) (auto simp: measure_nonneg)
hoelzl@50419
   276
  also have "\<dots> = ereal (measure lborel (A \<inter> {..a}) / r)"
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   277
    unfolding emeasure_eq_ereal_measure[OF fin] using A by (simp add: measure_def)
wenzelm@53015
   278
  finally show "(\<integral>\<^sup>+ x. ereal (indicator A x / measure lborel A * indicator {..a} x) \<partial>lborel) =
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   279
    ereal (measure lborel (A \<inter> {..a}) / r)" .
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   280
qed (auto simp: measure_nonneg)
hoelzl@50419
   281
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   282
lemma (in prob_space) uniform_distrI_borel_atLeastAtMost:
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   283
  fixes a b :: real
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   284
  assumes X: "X \<in> borel_measurable M" and "a < b"
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   285
  assumes distr: "\<And>t. a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow> \<P>(x in M. X x \<le> t) = (t - a) / (b - a)"
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   286
  shows "distributed M lborel X (\<lambda>x. indicator {a..b} x / measure lborel {a..b})"
hoelzl@50419
   287
proof (rule uniform_distrI_borel)
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   288
  fix t
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   289
  have "t < a \<or> (a \<le> t \<and> t \<le> b) \<or> b < t"
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   290
    by auto
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   291
  then show "emeasure M {x\<in>space M. X x \<le> t} = emeasure lborel ({a .. b} \<inter> {..t}) / (b - a)"
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   292
  proof (elim disjE conjE)
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   293
    assume "t < a" 
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   294
    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
   295
      using X by (auto intro!: emeasure_mono measurable_sets)
hoelzl@50419
   296
    also have "\<dots> = 0"
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   297
      using distr[of a] `a < b` by (simp add: emeasure_eq_measure)
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   298
    finally have "emeasure M {x\<in>space M. X x \<le> t} = 0"
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   299
      by (simp add: antisym measure_nonneg emeasure_le_0_iff)
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   300
    with `t < a` show ?thesis by simp
hoelzl@50419
   301
  next
hoelzl@50419
   302
    assume bnds: "a \<le> t" "t \<le> b"
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   303
    have "{a..b} \<inter> {..t} = {a..t}"
hoelzl@50419
   304
      using bnds by auto
hoelzl@50419
   305
    then show ?thesis using `a \<le> t` `a < b`
hoelzl@50419
   306
      using distr[OF bnds] by (simp add: emeasure_eq_measure)
hoelzl@50419
   307
  next
hoelzl@50419
   308
    assume "b < t" 
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   309
    have "1 = emeasure M {x\<in>space M. X x \<le> b}"
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   310
      using distr[of b] `a < b` by (simp add: one_ereal_def emeasure_eq_measure)
hoelzl@50419
   311
    also have "\<dots> \<le> emeasure M {x\<in>space M. X x \<le> t}"
hoelzl@50419
   312
      using X `b < t` by (auto intro!: emeasure_mono measurable_sets)
hoelzl@50419
   313
    finally have "emeasure M {x\<in>space M. X x \<le> t} = 1"
hoelzl@50419
   314
       by (simp add: antisym emeasure_eq_measure one_ereal_def)
hoelzl@50419
   315
    with `b < t` `a < b` show ?thesis by (simp add: measure_def one_ereal_def)
hoelzl@50419
   316
  qed
hoelzl@50419
   317
qed (insert X `a < b`, auto)
hoelzl@50419
   318
hoelzl@50419
   319
lemma (in prob_space) uniform_distributed_measure:
hoelzl@50419
   320
  fixes a b :: real
hoelzl@50419
   321
  assumes D: "distributed M lborel X (\<lambda>x. indicator {a .. b} x / measure lborel {a .. b})"
hoelzl@50419
   322
  assumes " a \<le> t" "t \<le> b"
hoelzl@50419
   323
  shows "\<P>(x in M. X x \<le> t) = (t - a) / (b - a)"
hoelzl@50419
   324
proof -
hoelzl@50419
   325
  have "emeasure M {x \<in> space M. X x \<le> t} = emeasure (distr M lborel X) {.. t}"
hoelzl@50419
   326
    using distributed_measurable[OF D]
hoelzl@50419
   327
    by (subst emeasure_distr) (auto intro!: arg_cong2[where f=emeasure])
wenzelm@53015
   328
  also have "\<dots> = (\<integral>\<^sup>+x. ereal (1 / (b - a)) * indicator {a .. t} x \<partial>lborel)"
hoelzl@50419
   329
    using distributed_borel_measurable[OF D] `a \<le> t` `t \<le> b`
hoelzl@50419
   330
    unfolding distributed_distr_eq_density[OF D]
hoelzl@50419
   331
    by (subst emeasure_density)
hoelzl@50419
   332
       (auto intro!: positive_integral_cong simp: measure_def split: split_indicator)
hoelzl@50419
   333
  also have "\<dots> = ereal (1 / (b - a)) * (t - a)"
hoelzl@50419
   334
    using `a \<le> t` `t \<le> b`
hoelzl@50419
   335
    by (subst positive_integral_cmult_indicator) auto
hoelzl@50419
   336
  finally show ?thesis
hoelzl@50419
   337
    by (simp add: measure_def)
hoelzl@50419
   338
qed
hoelzl@50419
   339
hoelzl@50419
   340
lemma (in prob_space) uniform_distributed_bounds:
hoelzl@50419
   341
  fixes a b :: real
hoelzl@50419
   342
  assumes D: "distributed M lborel X (\<lambda>x. indicator {a .. b} x / measure lborel {a .. b})"
hoelzl@50419
   343
  shows "a < b"
hoelzl@50419
   344
proof (rule ccontr)
hoelzl@50419
   345
  assume "\<not> a < b"
hoelzl@50419
   346
  then have "{a .. b} = {} \<or> {a .. b} = {a .. a}" by simp
hoelzl@50419
   347
  with uniform_distributed_params[OF D] show False 
hoelzl@50419
   348
    by (auto simp: measure_def)
hoelzl@50419
   349
qed
hoelzl@50419
   350
hoelzl@50419
   351
lemma (in prob_space) uniform_distributed_iff:
hoelzl@50419
   352
  fixes a b :: real
hoelzl@50419
   353
  shows "distributed M lborel X (\<lambda>x. indicator {a..b} x / measure lborel {a..b}) \<longleftrightarrow> 
hoelzl@50419
   354
    (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
   355
  using
hoelzl@50419
   356
    uniform_distributed_bounds[of X a b]
hoelzl@50419
   357
    uniform_distributed_measure[of X a b]
hoelzl@50419
   358
    distributed_measurable[of M lborel X]
hoelzl@50419
   359
  by (auto intro!: uniform_distrI_borel_atLeastAtMost simp: one_ereal_def emeasure_eq_measure)
hoelzl@50419
   360
hoelzl@50419
   361
lemma (in prob_space) uniform_distributed_expectation:
hoelzl@50419
   362
  fixes a b :: real
hoelzl@50419
   363
  assumes D: "distributed M lborel X (\<lambda>x. indicator {a .. b} x / measure lborel {a .. b})"
hoelzl@50419
   364
  shows "expectation X = (a + b) / 2"
hoelzl@50419
   365
proof (subst distributed_integral[OF D, of "\<lambda>x. x", symmetric])
hoelzl@50419
   366
  have "a < b"
hoelzl@50419
   367
    using uniform_distributed_bounds[OF D] .
hoelzl@50419
   368
hoelzl@50419
   369
  have "(\<integral> x. indicator {a .. b} x / measure lborel {a .. b} * x \<partial>lborel) = 
hoelzl@50419
   370
    (\<integral> x. (x / measure lborel {a .. b}) * indicator {a .. b} x \<partial>lborel)"
hoelzl@50419
   371
    by (intro integral_cong) auto
hoelzl@50419
   372
  also have "(\<integral> x. (x / measure lborel {a .. b}) * indicator {a .. b} x \<partial>lborel) = (a + b) / 2"
hoelzl@50419
   373
  proof (subst integral_FTC_atLeastAtMost)
hoelzl@50419
   374
    fix x
wenzelm@53077
   375
    show "DERIV (\<lambda>x. x\<^sup>2 / (2 * measure lborel {a..b})) x :> x / measure lborel {a..b}"
hoelzl@50419
   376
      using uniform_distributed_params[OF D]
hoelzl@56381
   377
      by (auto intro!: derivative_eq_intros)
hoelzl@50419
   378
    show "isCont (\<lambda>x. x / Sigma_Algebra.measure lborel {a..b}) x"
hoelzl@50419
   379
      using uniform_distributed_params[OF D]
hoelzl@50419
   380
      by (auto intro!: isCont_divide)
wenzelm@53015
   381
    have *: "b\<^sup>2 / (2 * measure lborel {a..b}) - a\<^sup>2 / (2 * measure lborel {a..b}) =
hoelzl@50419
   382
      (b*b - a * a) / (2 * (b - a))"
hoelzl@50419
   383
      using `a < b`
hoelzl@50419
   384
      by (auto simp: measure_def power2_eq_square diff_divide_distrib[symmetric])
wenzelm@53015
   385
    show "b\<^sup>2 / (2 * measure lborel {a..b}) - a\<^sup>2 / (2 * measure lborel {a..b}) = (a + b) / 2"
hoelzl@50419
   386
      using `a < b`
hoelzl@50419
   387
      unfolding * square_diff_square_factored by (auto simp: field_simps)
hoelzl@50419
   388
  qed (insert `a < b`, simp)
hoelzl@50419
   389
  finally show "(\<integral> x. indicator {a .. b} x / measure lborel {a .. b} * x \<partial>lborel) = (a + b) / 2" .
hoelzl@50419
   390
qed auto
hoelzl@50419
   391
hoelzl@50419
   392
end