section ‹Properties of Various Distributions›
theory Distributions
imports Convolution Information
begin
lemma (in prob_space) distributed_affine:
fixes f :: "real ⇒ ennreal"
assumes f: "distributed M lborel X f"
assumes c: "c ≠ 0"
shows "distributed M lborel (λx. t + c * X x) (λx. f ((x - t) / c) / ¦c¦)"
unfolding distributed_def
proof safe
have [measurable]: "f ∈ borel_measurable borel"
using f by (simp add: distributed_def)
have [measurable]: "X ∈ borel_measurable M"
using f by (simp add: distributed_def)
show "(λx. f ((x - t) / c) / ¦c¦) ∈ borel_measurable lborel"
by simp
show "random_variable lborel (λx. t + c * X x)"
by simp
have eq: "ennreal ¦c¦ * (f x / ennreal ¦c¦) = f x" for x
using c
by (cases "f x")
(auto simp: divide_ennreal ennreal_mult[symmetric] ennreal_top_divide ennreal_mult_top)
have "density lborel f = distr M lborel X"
using f by (simp add: distributed_def)
with c show "distr M lborel (λx. t + c * X x) = density lborel (λx. f ((x - t) / c) / ennreal ¦c¦)"
by (subst (2) lborel_real_affine[where c="c" and t="t"])
(simp_all add: density_density_eq density_distr distr_distr field_simps eq cong: distr_cong)
qed
lemma (in prob_space) distributed_affineI:
fixes f :: "real ⇒ ennreal" and c :: real
assumes f: "distributed M lborel (λx. (X x - t) / c) (λx. ¦c¦ * f (x * c + t))"
assumes c: "c ≠ 0"
shows "distributed M lborel X f"
proof -
have eq: "f x * ennreal ¦c¦ / ennreal ¦c¦ = f x" for x
using c by (simp add: ennreal_times_divide[symmetric])
show ?thesis
using distributed_affine[OF f c, where t=t] c
by (simp add: field_simps eq)
qed
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)) ⟷ (AE x in N. 0 < f x ⟶ P x)"
proof -
have "(AE x in M. P (X x)) ⟷ (AE x in distr M N X. P x)"
by (simp add: AE_distr_iff)
also have "… ⟷ (AE x in density N f. P x)"
unfolding distributed_distr_eq_density[OF assms(1)] ..
also have "… ⟷ (AE x in N. 0 < f x ⟶ P x)"
by (rule AE_density) simp
finally show ?thesis .
qed
subsection ‹Erlang›
lemma nn_intergal_power_times_exp_Icc:
assumes [arith]: "0 ≤ a"
shows "(∫⇧+x. ennreal (x^k * exp (-x)) * indicator {0 .. a} x ∂lborel) =
(1 - (∑n≤k. (a^n * exp (-a)) / fact n)) * fact k" (is "?I = _")
proof -
let ?f = "λk x. x^k * exp (-x) / fact k"
let ?F = "λk x. - (∑n≤k. (x^n * exp (-x)) / fact n)"
have "?I * (inverse (real_of_nat (fact k))) =
(∫⇧+x. ennreal (x^k * exp (-x)) * indicator {0 .. a} x * (inverse (real_of_nat (fact k))) ∂lborel)"
by (intro nn_integral_multc[symmetric]) auto
also have "… = (∫⇧+x. ennreal (?f k x) * indicator {0 .. a} x ∂lborel)"
by (intro nn_integral_cong)
(simp add: field_simps ennreal_mult'[symmetric] indicator_mult_ennreal)
also have "… = ennreal (?F k a - ?F k 0)"
proof (rule nn_integral_FTC_Icc)
fix x assume "x ∈ {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 (λx. ?F k x - (x^Suc k * exp (-x)) / fact (Suc k)) x :>
?f k x - ((real (Suc k) - x) * x ^ k * exp (- x)) / (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])
also have "(λx. ?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)) / (fact (Suc k)) = ?f (Suc k) x"
by (auto simp: field_simps simp del: fact_Suc)
(simp_all add: of_nat_Suc field_simps)
finally show ?case .
qed
qed auto
also have "… = ennreal (1 - (∑n≤k. (a^n * exp (-a)) / fact n))"
by (auto simp: power_0_left if_distrib[where f="λx. x / a" for a] setsum.If_cases)
also have "… = ennreal ((1 - (∑n≤k. (a^n * exp (-a)) / fact n)) * fact k) * ennreal (inverse (fact k))"
by (subst ennreal_mult''[symmetric]) (auto intro!: arg_cong[where f=ennreal])
finally show ?thesis
by (auto simp add: mult_right_ennreal_cancel le_less)
qed
lemma nn_intergal_power_times_exp_Ici:
shows "(∫⇧+x. ennreal (x^k * exp (-x)) * indicator {0 ..} x ∂lborel) = real_of_nat (fact k)"
proof (rule LIMSEQ_unique)
let ?X = "λn. ∫⇧+ x. ennreal (x^k * exp (-x)) * indicator {0 .. real n} x ∂lborel"
show "?X ⇢ (∫⇧+x. ennreal (x^k * exp (-x)) * indicator {0 ..} x ∂lborel)"
apply (intro nn_integral_LIMSEQ)
apply (auto simp: incseq_def le_fun_def eventually_sequentially
split: split_indicator intro!: Lim_eventually)
apply (metis nat_ceiling_le_eq)
done
have "((λx::real. (1 - (∑n≤k. (x ^ n / exp x) / (fact n))) * fact k) ⤏
(1 - (∑n≤k. 0 / (fact n))) * fact k) at_top"
by (intro tendsto_intros tendsto_power_div_exp_0) simp
then show "?X ⇢ real_of_nat (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 ⇒ real ⇒ real ⇒ 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 ⇒ real ⇒ real ⇒ real" where
"erlang_CDF k l x = (if x < 0 then 0 else 1 - (∑n≤k. ((l * x)^n * exp (- l * x) / fact n)))"
lemma erlang_density_nonneg[simp]: "0 ≤ l ⟹ 0 ≤ erlang_density k l x"
by (simp add: erlang_density_def)
lemma borel_measurable_erlang_density[measurable]: "erlang_density k l ∈ borel_measurable borel"
by (auto simp add: erlang_density_def[abs_def])
lemma erlang_CDF_transform: "0 < l ⟹ erlang_CDF k l a = erlang_CDF k 1 (l * a)"
by (auto simp add: erlang_CDF_def mult_less_0_iff)
lemma erlang_CDF_nonneg[simp]: assumes "0 < l" shows "0 ≤ erlang_CDF k l x"
unfolding erlang_CDF_def
proof (clarsimp simp: not_less)
assume "0 ≤ x"
have "(∑n≤k. (l * x) ^ n * exp (- (l * x)) / fact n) =
exp (- (l * x)) * (∑n≤k. (l * x) ^ n / fact n)"
unfolding setsum_right_distrib by (intro setsum.cong) (auto simp: field_simps)
also have "… = (∑n≤k. (l * x) ^ n / fact n) / exp (l * x)"
by (simp add: exp_minus field_simps)
also have "… ≤ 1"
proof (subst divide_le_eq_1_pos)
show "(∑n≤k. (l * x) ^ n / fact n) ≤ exp (l * x)"
using ‹0 < l› ‹0 ≤ x› summable_exp_generic[of "l * x"]
by (auto simp: exp_def divide_inverse ac_simps intro!: setsum_le_suminf)
qed simp
finally show "(∑n≤k. (l * x) ^ n * exp (- (l * x)) / fact n) ≤ 1" .
qed
lemma nn_integral_erlang_density:
assumes [arith]: "0 < l"
shows "(∫⇧+ x. ennreal (erlang_density k l x) * indicator {.. a} x ∂lborel) = erlang_CDF k l a"
proof cases
assume [arith]: "0 ≤ a"
have eq: "⋀x. indicator {0..a} (x / l) = indicator {0..a*l} x"
by (simp add: field_simps split: split_indicator)
have "(∫⇧+x. ennreal (erlang_density k l x) * indicator {.. a} x ∂lborel) =
(∫⇧+x. (l/fact k) * (ennreal ((l*x)^k * exp (- (l*x))) * indicator {0 .. a} x) ∂lborel)"
by (intro nn_integral_cong)
(auto simp: erlang_density_def power_mult_distrib ennreal_mult[symmetric] split: split_indicator)
also have "… = (l/fact k) * (∫⇧+x. ennreal ((l*x)^k * exp (- (l*x))) * indicator {0 .. a} x ∂lborel)"
by (intro nn_integral_cmult) auto
also have "… = ennreal (l/fact k) * ((1/l) * (∫⇧+x. ennreal (x^k * exp (- x)) * indicator {0 .. l * a} x ∂lborel))"
by (subst nn_integral_real_affine[where c="1 / l" and t=0]) (auto simp: field_simps eq)
also have "… = (1 - (∑n≤k. ((l * a)^n * exp (-(l * a))) / fact n))"
by (subst nn_intergal_power_times_exp_Icc) (auto simp: ennreal_mult'[symmetric])
also have "… = erlang_CDF k l a"
by (auto simp: erlang_CDF_def)
finally show ?thesis .
next
assume "¬ 0 ≤ a"
moreover then have "(∫⇧+ x. ennreal (erlang_density k l x) * indicator {.. a} x ∂lborel) = (∫⇧+x. 0 ∂(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 ⟹ 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 "(∫⇧+ x. ennreal (erlang_density k l x * x ^ i) ∂lborel) = fact (k + i) / (fact k * l ^ i)"
proof -
have eq: "⋀x. indicator {0..} (x / l) = indicator {0..} x"
by (simp add: field_simps split: split_indicator)
have "(∫⇧+ x. ennreal (erlang_density k l x * x^i) ∂lborel) =
(∫⇧+x. (l/(fact k * l^i)) * (ennreal ((l*x)^(k+i) * exp (- (l*x))) * indicator {0 ..} x) ∂lborel)"
by (intro nn_integral_cong)
(auto simp: erlang_density_def power_mult_distrib power_add ennreal_mult'[symmetric] split: split_indicator)
also have "… = (l/(fact k * l^i)) * (∫⇧+x. ennreal ((l*x)^(k+i) * exp (- (l*x))) * indicator {0 ..} x ∂lborel)"
by (intro nn_integral_cmult) auto
also have "… = ennreal (l/(fact k * l^i)) * ((1/l) * (∫⇧+x. ennreal (x^(k+i) * exp (- x)) * indicator {0 ..} x ∂lborel))"
by (subst nn_integral_real_affine[where c="1 / l" and t=0]) (auto simp: field_simps eq)
also have "… = fact (k + i) / (fact k * l ^ i)"
by (subst nn_intergal_power_times_exp_Ici) (auto simp: ennreal_mult'[symmetric])
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 ≤ a"
shows "𝒫(x in M. X x ≤ a) = erlang_CDF k l a"
proof -
have "emeasure M {x ∈ space M. X x ≤ 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 "… = emeasure (density lborel (erlang_density k l)) {.. a}"
unfolding distributed_distr_eq_density[OF D] ..
also have "… = erlang_CDF k l a"
by (auto intro!: emeasure_erlang_density)
finally show ?thesis
by (auto simp: emeasure_eq_measure measure_nonneg)
qed
lemma (in prob_space) erlang_distributed_gt:
assumes D[simp]: "distributed M lborel X (erlang_density k l)"
assumes [arith]: "0 < l" "0 ≤ 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 ≤ a)" by (subst erlang_distributed_le) auto
also have "… = prob (space M - {x ∈ space M. X x ≤ a })"
using distributed_measurable[OF D] by (auto simp: prob_compl)
also have "… = 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 ∈ borel_measurable M" and [arith]: "0 < l"
and X_distr: "⋀a. 0 ≤ a ⟹ emeasure M {x∈space M. X x ≤ 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 ≤ 0"
with X have "emeasure M {x∈space M. X x ≤ a} ≤ emeasure M {x∈space M. X x ≤ 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∈space M. X x ≤ a} ≤ 0" by simp
then have "emeasure M {x∈space M. X x ≤ a} = 0" by simp
}
note eq_0 = this
show "(∫⇧+ x. erlang_density k l x * indicator {..a} x ∂lborel) = ennreal (erlang_CDF k l a)"
using nn_integral_erlang_density[of l k a]
by (simp add: ennreal_indicator ennreal_mult)
show "emeasure M {x∈space M. X x ≤ a} = ennreal (erlang_CDF k l a)"
using X_distr[of a] eq_0 by (auto simp: one_ennreal_def erlang_CDF_def)
qed simp_all
lemma (in prob_space) erlang_distributed_iff:
assumes [arith]: "0<l"
shows "distributed M lborel X (erlang_density k l) ⟷
(X ∈ borel_measurable M ∧ 0 < l ∧ (∀a≥0. 𝒫(x in M. X x ≤ 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_ennreal_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 < α" "0 < l"
shows "distributed M lborel (λx. α * X x) (erlang_density k (l / α))"
proof (subst erlang_distributed_iff, safe)
have [measurable]: "random_variable borel X" and [arith]: "0 < l "
and [simp]: "⋀a. 0 ≤ a ⟹ prob {x ∈ space M. X x ≤ a} = erlang_CDF k l a"
by(insert erlX, auto simp: erlang_distributed_iff)
show "random_variable borel (λx. α * X x)" "0 < l / α" "0 < l / α"
by (auto simp:field_simps)
fix a:: real assume [arith]: "0 ≤ a"
obtain b:: real where [simp, arith]: "b = a/ α" by blast
have [arith]: "0 ≤ b" by (auto simp: divide_nonneg_pos)
have "prob {x ∈ space M. α * X x ≤ a} = prob {x ∈ space M. X x ≤ b}"
by (rule arg_cong[where f= prob]) (auto simp:field_simps)
moreover have "prob {x ∈ space M. X x ≤ b} = erlang_CDF k l b" by auto
moreover have "erlang_CDF k (l / α) a = erlang_CDF k l b" unfolding erlang_CDF_def by auto
ultimately show "prob {x ∈ space M. α * X x ≤ a} = erlang_CDF k (l / α) 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 (λx. X x ^ i) (fact (k + i) / (fact k * l ^ i))"
proof (rule has_bochner_integral_nn_integral)
show "AE x in M. 0 ≤ X x ^ i"
by (subst distributed_AE2[OF D]) (auto simp: erlang_density_def)
show "(∫⇧+ x. ennreal (X x ^ i) ∂M) = ennreal (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 simp: ennreal_mult')
qed (insert distributed_measurable[OF D], auto)
lemma (in prob_space) erlang_ith_moment_integrable:
"0 < l ⟹ distributed M lborel X (erlang_density k l) ⟹ integrable M (λx. X x ^ i)"
by rule (rule has_bochner_integral_erlang_ith_moment)
lemma (in prob_space) erlang_ith_moment:
"0 < l ⟹ distributed M lborel X (erlang_density k l) ⟹
expectation (λx. 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⇧2"
proof (subst variance_eq)
show "integrable M X" "integrable M (λx. (X x)⇧2)"
using erlang_ith_moment_integrable[OF assms, of 1] erlang_ith_moment_integrable[OF assms, of 2]
by auto
show "expectation (λx. (X x)⇧2) - (expectation X)⇧2 = real (k + 1) / l⇧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 of_nat_Suc)
qed
subsection ‹Exponential distribution›
abbreviation exponential_density :: "real ⇒ real ⇒ real" where
"exponential_density ≡ erlang_density 0"
lemma exponential_density_def:
"exponential_density l x = (if x < 0 then 0 else l * exp (- x * l))"
by (simp add: fun_eq_iff erlang_density_def)
lemma erlang_CDF_0: "erlang_CDF 0 l a = (if 0 ≤ a then 1 - exp (- l * a) else 0)"
by (simp add: erlang_CDF_def)
lemma prob_space_exponential_density: "0 < l ⟹ 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 ≤ a" and l: "0 < l"
shows "𝒫(x in M. X x ≤ a) = 1 - exp (- a * l)"
using erlang_distributed_le[OF D l 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 ≤ a" and l: "0 < l"
shows "𝒫(x in M. a < X x ) = exp (- a * l)"
using erlang_distributed_gt[OF D l 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 ≤ a" and l: "0 < l" and t: "0 ≤ t"
shows "𝒫(x in M. a + t < X x ¦ a < X x) = 𝒫(x in M. t < X x)"
proof -
have "𝒫(x in M. a + t < X x ¦ a < X x) = 𝒫(x in M. a + t < X x) / 𝒫(x in M. a < X x)"
using ‹0 ≤ t› by (auto simp: cond_prob_def intro!: arg_cong[where f=prob] arg_cong2[where f="op /"])
also have "… = exp (- (a + t) * l) / exp (- a * l)"
using a t by (simp add: exponential_distributedD_gt[OF D _ l])
also have "… = exp (- t * l)"
using l by (auto simp: field_simps exp_add[symmetric])
finally show ?thesis
using t by (simp add: exponential_distributedD_gt[OF D _ l])
qed
lemma exponential_distributedI:
assumes X[measurable]: "X ∈ borel_measurable M" and [arith]: "0 < l"
and X_distr: "⋀a. 0 ≤ a ⟹ emeasure M {x∈space M. X x ≤ a} = 1 - exp (- a * l)"
shows "distributed M lborel X (exponential_density l)"
proof (rule erlang_distributedI)
fix a :: real assume "0 ≤ a" then show "emeasure M {x ∈ space M. X x ≤ a} = ennreal (erlang_CDF 0 l a)"
using X_distr[of a] by (simp add: erlang_CDF_def ennreal_minus ennreal_1[symmetric] del: ennreal_1)
qed fact+
lemma (in prob_space) exponential_distributed_iff:
assumes "0 < l"
shows "distributed M lborel X (exponential_density l) ⟷
(X ∈ borel_measurable M ∧ (∀a≥0. 𝒫(x in M. X x ≤ a) = 1 - exp (- a * l)))"
using assms erlang_distributed_iff[of l X 0] by (auto simp: erlang_CDF_0)
lemma (in prob_space) exponential_distributed_expectation:
"0 < l ⟹ distributed M lborel X (exponential_density l) ⟹ expectation X = 1 / l"
using erlang_ith_moment[of l X 0 1] by simp
lemma exponential_density_nonneg: "0 < l ⟹ 0 ≤ exponential_density l x"
by (auto simp: exponential_density_def)
lemma (in prob_space) exponential_distributed_min:
assumes "0 < l" "0 < u"
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 (λx. min (X x) (Y x)) (exponential_density (l + u))"
proof (subst exponential_distributed_iff, safe)
have randX: "random_variable borel X"
using expX ‹0 < l› by (simp add: exponential_distributed_iff)
moreover have randY: "random_variable borel Y"
using expY ‹0 < u› by (simp add: exponential_distributed_iff)
ultimately show "random_variable borel (λx. min (X x) (Y x))" by auto
show " 0 < l + u"
using ‹0 < l› ‹0 < u› by auto
fix a::real assume a[arith]: "0 ≤ a"
have gt1[simp]: "𝒫(x in M. a < X x ) = exp (- a * l)"
by (rule exponential_distributedD_gt[OF expX a]) fact
have gt2[simp]: "𝒫(x in M. a < Y x ) = exp (- a * u)"
by (rule exponential_distributedD_gt[OF expY a]) fact
have "𝒫(x in M. a < (min (X x) (Y x)) ) = 𝒫(x in M. a < (X x) ∧ 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 ∈ space M. (min (X x) (Y x)) ≤a } = (space M - {x ∈ space M. a<(min (X x) (Y x)) })"
by auto
then have "1 - prob {x ∈ space M. a < (min (X x) (Y x))} = prob {x ∈ space M. (min (X x) (Y x)) ≤ a}"
using randX randY by (auto simp: prob_compl)
then show "prob {x ∈ space M. (min (X x) (Y x)) ≤ 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 ≠ {}"
assumes l: "⋀i. i ∈ I ⟹ 0 < l i"
assumes expX: "⋀i. i ∈ I ⟹ distributed M lborel (X i) (exponential_density (l i))"
assumes ind: "indep_vars (λi. borel) X I"
shows "distributed M lborel (λx. Min ((λi. X i x)`I)) (exponential_density (∑i∈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 (λx. min (X i x) (Min ((λi. X i x)`I))) (exponential_density (l i + (∑i∈I. l i)))"
by (intro exponential_distributed_min indep_vars_Min insert)
(auto intro: indep_vars_subset setsum_pos)
then show ?case
using insert by simp
qed
lemma (in prob_space) exponential_distributed_variance:
"0 < l ⟹ distributed M lborel X (exponential_density l) ⟹ variance X = 1 / l⇧2"
using erlang_distributed_variance[of l X 0] by simp
lemma nn_integral_zero': "AE x in M. f x = 0 ⟹ (∫⇧+x. f x ∂M) = 0"
by (simp cong: nn_integral_cong_AE)
lemma convolution_erlang_density:
fixes k⇩1 k⇩2 :: nat
assumes [simp, arith]: "0 < l"
shows "(λx. ∫⇧+y. ennreal (erlang_density k⇩1 l (x - y)) * ennreal (erlang_density k⇩2 l y) ∂lborel) =
(erlang_density (Suc k⇩1 + Suc k⇩2 - 1) l)"
(is "?LHS = ?RHS")
proof
fix x :: real
have "x ≤ 0 ∨ 0 < x"
by arith
then show "?LHS x = ?RHS x"
proof
assume "x ≤ 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⇧N lborel (erlang_density (Suc k⇩1 + Suc k⇩2 - 1) l) = 1"
using nn_integral_erlang_ith_moment[of l "Suc k⇩1 + Suc k⇩2 - 1" 0] by (simp del: fact_Suc)
have 1: "(∫⇧+ x. ennreal (erlang_density (Suc k⇩1 + Suc k⇩2 - 1) l x * indicator {0<..} x) ∂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)"
by (intro prob_space_convolution_density)
(auto intro!: prob_space_erlang_density erlang_density_nonneg)
then have 2: "integral⇧N lborel ?LHS = 1"
by (auto dest!: prob_space.emeasure_space_1 simp: emeasure_density)
let ?I = "(integral⇧N lborel (λy. ennreal ((1 - y)^ k⇩1 * y^k⇩2 * indicator {0..1} y)))"
let ?C = "(fact (Suc (k⇩1 + k⇩2))) / ((fact k⇩1) * (fact k⇩2))"
let ?s = "Suc k⇩1 + Suc k⇩2 - 1"
let ?L = "(λx. ∫⇧+y. ennreal (erlang_density k⇩1 l (x- y) * erlang_density k⇩2 l y * indicator {0..x} y) ∂lborel)"
{ fix x :: real assume [arith]: "0 < x"
have *: "⋀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 simp: ennreal_mult split:split_indicator)
also have "... = (λx. ennreal ?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] 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 *
ennreal_mult[symmetric]
simp del: fact_Suc split: split_indicator)
done
finally have "(∫⇧+y. ennreal (erlang_density k⇩1 l (x - y) * erlang_density k⇩2 l y) ∂lborel) =
(λx. ennreal ?C * ?I * erlang_density ?s l x) x"
by (simp add: ennreal_mult) }
note * = this
assume [arith]: "0 < x"
have 3: "1 = integral⇧N lborel (λxa. ?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: if_split_asm)
also have "... = integral⇧N lborel (λx. (ennreal (?C) * ?I) * ((erlang_density ?s l x) * indicator {0<..} x))"
by (auto intro!: nn_integral_cong simp: ennreal_mult[symmetric] * split: split_indicator)
also have "... = ennreal (?C) * ?I"
using 1
by (auto simp: nn_integral_cmult)
finally have " ennreal (?C) * ?I = 1" by presburger
then show ?thesis
using * by (simp add: ennreal_mult)
qed
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⇩1 l)"
assumes erlY: "distributed M lborel Y (erlang_density k⇩2 l)"
shows "distributed M lborel (λx. X x + Y x) (erlang_density (Suc k⇩1 + Suc k⇩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 ≠ {}"
assumes [simp, arith]: "0 < l"
assumes expX: "⋀i. i ∈ I ⟹ distributed M lborel (X i) (erlang_density (k i) l)"
assumes ind: "indep_vars (λi. borel) X I"
shows "distributed M lborel (λx. ∑i∈I. X i x) (erlang_density ((∑i∈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 (λx. (X i x) + (∑i∈ I. X i x)) (erlang_density (Suc (k i) + Suc ((∑i∈I. Suc (k i)) - 1) - 1) l)"
by(intro sum_indep_erlang indep_vars_setsum) (auto intro!: indep_vars_subset)
also have "(λx. (X i x) + (∑i∈ I. X i x)) = (λx. ∑i∈insert i I. X i x)"
using insert by auto
also have "Suc(k i) + Suc ((∑i∈I. Suc (k i)) - 1) - 1 = (∑i∈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 ≠ {}"
assumes l: "0 < l"
assumes expX: "⋀i. i ∈ I ⟹ distributed M lborel (X i) (exponential_density l)"
assumes ind: "indep_vars (λi. borel) X I"
shows "distributed M lborel (λx. ∑i∈I. X i x) (erlang_density ((card I) - 1) l)"
using erlang_distributed_setsum[OF assms] by simp
lemma (in information_space) entropy_exponential:
assumes l[simp, arith]: "0 < l"
assumes D: "distributed M lborel X (exponential_density l)"
shows "entropy b lborel X = log b (exp 1 / l)"
proof -
have [simp]: "integrable lborel (exponential_density l)"
using distributed_integrable[OF D, of "λ_. 1"] by simp
have [simp]: "integral⇧L lborel (exponential_density l) = 1"
using distributed_integral[OF D, of "λ_. 1"] by (simp add: prob_space)
have [simp]: "integrable lborel (λx. exponential_density l x * x)"
using erlang_ith_moment_integrable[OF l D, of 1] distributed_integrable[OF D, of "λx. x"] by simp
have [simp]: "integral⇧L lborel (λx. exponential_density l x * x) = 1 / l"
using erlang_ith_moment[OF l D, of 1] distributed_integral[OF D, of "λx. x"] by simp
have "entropy b lborel X = - (∫ x. exponential_density l x * log b (exponential_density l x) ∂lborel)"
using D by (rule entropy_distr) simp
also have "(∫ x. exponential_density l x * log b (exponential_density l x) ∂lborel) =
(∫ x. (ln l * exponential_density l x - l * (exponential_density l x * x)) / ln b ∂lborel)"
by (intro integral_cong) (auto simp: log_def ln_mult exponential_density_def field_simps)
also have "… = (ln l - 1) / ln b"
by simp
finally show ?thesis
by (simp add: log_def divide_simps ln_div)
qed
subsection ‹Uniform distribution›
lemma uniform_distrI:
assumes X: "X ∈ measurable M M'"
and A: "A ∈ sets M'" "emeasure M' A ≠ ∞" "emeasure M' A ≠ 0"
assumes distr: "⋀B. B ∈ sets M' ⟹ emeasure M (X -` B ∩ space M) = emeasure M' (A ∩ B) / emeasure M' A"
shows "distr M M' X = uniform_measure M' A"
unfolding uniform_measure_def
proof (intro measure_eqI)
let ?f = "λx. indicator A x / emeasure M' A"
fix B assume B: "B ∈ sets (distr M M' X)"
with X have "emeasure M (X -` B ∩ space M) = emeasure M' (A ∩ B) / emeasure M' A"
by (simp add: distr[of B] measurable_sets)
also have "… = (1 / emeasure M' A) * emeasure M' (A ∩ B)"
by (simp add: divide_ennreal_def ac_simps)
also have "… = (∫⇧+ x. (1 / emeasure M' A) * indicator (A ∩ B) x ∂M')"
using A B
by (intro nn_integral_cmult_indicator[symmetric]) (auto intro!: )
also have "… = (∫⇧+ x. ?f x * indicator B x ∂M')"
by (rule nn_integral_cong) (auto split: split_indicator)
finally show "emeasure (distr M M' X) B = emeasure (density M' ?f) B"
using A B X by (auto simp add: emeasure_distr emeasure_density)
qed simp
lemma uniform_distrI_borel:
fixes A :: "real set"
assumes X[measurable]: "X ∈ borel_measurable M" and A: "emeasure lborel A = ennreal r" "0 < r"
and [measurable]: "A ∈ sets borel"
assumes distr: "⋀a. emeasure M {x∈space M. X x ≤ a} = emeasure lborel (A ∩ {.. a}) / r"
shows "distributed M lborel X (λx. indicator A x / measure lborel A)"
proof (rule distributedI_borel_atMost)
let ?f = "λx. 1 / r * indicator A x"
fix a
have "emeasure lborel (A ∩ {..a}) ≤ emeasure lborel A"
using A by (intro emeasure_mono) auto
also have "… < ∞"
using A by simp
finally have fin: "emeasure lborel (A ∩ {..a}) ≠ top"
by simp
from emeasure_eq_ennreal_measure[OF this]
have fin_eq: "emeasure lborel (A ∩ {..a}) / r = ennreal (measure lborel (A ∩ {..a}) / r)"
using A by (simp add: divide_ennreal measure_nonneg)
then show "emeasure M {x∈space M. X x ≤ a} = ennreal (measure lborel (A ∩ {..a}) / r)"
using distr by simp
have "(∫⇧+ x. ennreal (indicator A x / measure lborel A * indicator {..a} x) ∂lborel) =
(∫⇧+ x. ennreal (1 / measure lborel A) * indicator (A ∩ {..a}) x ∂lborel)"
by (auto intro!: nn_integral_cong split: split_indicator)
also have "… = ennreal (1 / measure lborel A) * emeasure lborel (A ∩ {..a})"
using ‹A ∈ sets borel›
by (intro nn_integral_cmult_indicator) (auto simp: measure_nonneg)
also have "… = ennreal (measure lborel (A ∩ {..a}) / r)"
unfolding emeasure_eq_ennreal_measure[OF fin] using A
by (simp add: measure_def ennreal_mult'[symmetric])
finally show "(∫⇧+ x. ennreal (indicator A x / measure lborel A * indicator {..a} x) ∂lborel) =
ennreal (measure lborel (A ∩ {..a}) / r)" .
qed (auto simp: measure_nonneg)
lemma (in prob_space) uniform_distrI_borel_atLeastAtMost:
fixes a b :: real
assumes X: "X ∈ borel_measurable M" and "a < b"
assumes distr: "⋀t. a ≤ t ⟹ t ≤ b ⟹ 𝒫(x in M. X x ≤ t) = (t - a) / (b - a)"
shows "distributed M lborel X (λx. indicator {a..b} x / measure lborel {a..b})"
proof (rule uniform_distrI_borel)
fix t
have "t < a ∨ (a ≤ t ∧ t ≤ b) ∨ b < t"
by auto
then show "emeasure M {x∈space M. X x ≤ t} = emeasure lborel ({a .. b} ∩ {..t}) / (b - a)"
proof (elim disjE conjE)
assume "t < a"
then have "emeasure M {x∈space M. X x ≤ t} ≤ emeasure M {x∈space M. X x ≤ a}"
using X by (auto intro!: emeasure_mono measurable_sets)
also have "… = 0"
using distr[of a] ‹a < b› by (simp add: emeasure_eq_measure)
finally have "emeasure M {x∈space M. X x ≤ t} = 0"
by (simp add: antisym measure_nonneg)
with ‹t < a› show ?thesis by simp
next
assume bnds: "a ≤ t" "t ≤ b"
have "{a..b} ∩ {..t} = {a..t}"
using bnds by auto
then show ?thesis using ‹a ≤ t› ‹a < b›
using distr[OF bnds] by (simp add: emeasure_eq_measure divide_ennreal)
next
assume "b < t"
have "1 = emeasure M {x∈space M. X x ≤ b}"
using distr[of b] ‹a < b› by (simp add: one_ennreal_def emeasure_eq_measure)
also have "… ≤ emeasure M {x∈space M. X x ≤ t}"
using X ‹b < t› by (auto intro!: emeasure_mono measurable_sets)
finally have "emeasure M {x∈space M. X x ≤ t} = 1"
by (simp add: antisym emeasure_eq_measure)
with ‹b < t› ‹a < b› show ?thesis by (simp add: measure_def divide_ennreal)
qed
qed (insert X ‹a < b›, auto)
lemma (in prob_space) uniform_distributed_measure:
fixes a b :: real
assumes D: "distributed M lborel X (λx. indicator {a .. b} x / measure lborel {a .. b})"
assumes t: "a ≤ t" "t ≤ b"
shows "𝒫(x in M. X x ≤ t) = (t - a) / (b - a)"
proof -
have "emeasure M {x ∈ space M. X x ≤ t} = emeasure (distr M lborel X) {.. t}"
using distributed_measurable[OF D]
by (subst emeasure_distr) (auto intro!: arg_cong2[where f=emeasure])
also have "… = (∫⇧+x. ennreal (1 / (b - a)) * indicator {a .. t} x ∂lborel)"
using distributed_borel_measurable[OF D] ‹a ≤ t› ‹t ≤ b›
unfolding distributed_distr_eq_density[OF D]
by (subst emeasure_density)
(auto intro!: nn_integral_cong simp: measure_def split: split_indicator)
also have "… = ennreal (1 / (b - a)) * (t - a)"
using ‹a ≤ t› ‹t ≤ b›
by (subst nn_integral_cmult_indicator) auto
finally show ?thesis
using t by (simp add: emeasure_eq_measure ennreal_mult''[symmetric] measure_nonneg)
qed
lemma (in prob_space) uniform_distributed_bounds:
fixes a b :: real
assumes D: "distributed M lborel X (λx. indicator {a .. b} x / measure lborel {a .. b})"
shows "a < b"
proof (rule ccontr)
assume "¬ a < b"
then have "{a .. b} = {} ∨ {a .. b} = {a .. a}" by simp
with uniform_distributed_params[OF D] show False
by (auto simp: measure_def)
qed
lemma (in prob_space) uniform_distributed_iff:
fixes a b :: real
shows "distributed M lborel X (λx. indicator {a..b} x / measure lborel {a..b}) ⟷
(X ∈ borel_measurable M ∧ a < b ∧ (∀t∈{a .. b}. 𝒫(x in M. X x ≤ t)= (t - a) / (b - a)))"
using
uniform_distributed_bounds[of X a b]
uniform_distributed_measure[of X a b]
distributed_measurable[of M lborel X]
by (auto intro!: uniform_distrI_borel_atLeastAtMost)
lemma (in prob_space) uniform_distributed_expectation:
fixes a b :: real
assumes D: "distributed M lborel X (λx. indicator {a .. b} x / measure lborel {a .. b})"
shows "expectation X = (a + b) / 2"
proof (subst distributed_integral[OF D, of "λx. x", symmetric])
have "a < b"
using uniform_distributed_bounds[OF D] .
have "(∫ x. indicator {a .. b} x / measure lborel {a .. b} * x ∂lborel) =
(∫ x. (x / measure lborel {a .. b}) * indicator {a .. b} x ∂lborel)"
by (intro integral_cong) auto
also have "(∫ x. (x / measure lborel {a .. b}) * indicator {a .. b} x ∂lborel) = (a + b) / 2"
proof (subst integral_FTC_Icc_real)
fix x
show "DERIV (λx. x⇧2 / (2 * measure lborel {a..b})) x :> x / measure lborel {a..b}"
using uniform_distributed_params[OF D]
by (auto intro!: derivative_eq_intros)
show "isCont (λx. x / Sigma_Algebra.measure lborel {a..b}) x"
using uniform_distributed_params[OF D]
by (auto intro!: isCont_divide)
have *: "b⇧2 / (2 * measure lborel {a..b}) - a⇧2 / (2 * measure lborel {a..b}) =
(b*b - a * a) / (2 * (b - a))"
using ‹a < b›
by (auto simp: measure_def power2_eq_square diff_divide_distrib[symmetric])
show "b⇧2 / (2 * measure lborel {a..b}) - a⇧2 / (2 * measure lborel {a..b}) = (a + b) / 2"
using ‹a < b›
unfolding * square_diff_square_factored by (auto simp: field_simps)
qed (insert ‹a < b›, simp)
finally show "(∫ x. indicator {a .. b} x / measure lborel {a .. b} * x ∂lborel) = (a + b) / 2" .
qed (auto simp: measure_nonneg)
lemma (in prob_space) uniform_distributed_variance:
fixes a b :: real
assumes D: "distributed M lborel X (λx. indicator {a .. b} x / measure lborel {a .. b})"
shows "variance X = (b - a)⇧2 / 12"
proof (subst distributed_variance)
have [arith]: "a < b" using uniform_distributed_bounds[OF D] .
let ?μ = "expectation X" let ?D = "λx. indicator {a..b} (x + ?μ) / measure lborel {a..b}"
have "(∫x. x⇧2 * (?D x) ∂lborel) = (∫x. x⇧2 * (indicator {a - ?μ .. b - ?μ} x) / measure lborel {a .. b} ∂lborel)"
by (intro integral_cong) (auto split: split_indicator)
also have "… = (b - a)⇧2 / 12"
by (simp add: integral_power uniform_distributed_expectation[OF D])
(simp add: eval_nat_numeral field_simps )
finally show "(∫x. x⇧2 * ?D x ∂lborel) = (b - a)⇧2 / 12" .
qed (auto intro: D simp: measure_nonneg)
subsection ‹Normal distribution›
definition normal_density :: "real ⇒ real ⇒ real ⇒ real" where
"normal_density μ σ x = 1 / sqrt (2 * pi * σ⇧2) * exp (-(x - μ)⇧2/ (2 * σ⇧2))"
abbreviation std_normal_density :: "real ⇒ real" where
"std_normal_density ≡ normal_density 0 1"
lemma std_normal_density_def: "std_normal_density x = (1 / sqrt (2 * pi)) * exp (- x⇧2 / 2)"
unfolding normal_density_def by simp
lemma normal_density_nonneg[simp]: "0 ≤ normal_density μ σ x"
by (auto simp: normal_density_def)
lemma normal_density_pos: "0 < σ ⟹ 0 < normal_density μ σ x"
by (auto simp: normal_density_def)
lemma borel_measurable_normal_density[measurable]: "normal_density μ σ ∈ borel_measurable borel"
by (auto simp: normal_density_def[abs_def])
lemma gaussian_moment_0:
"has_bochner_integral lborel (λx. indicator {0..} x *⇩R exp (- x⇧2)) (sqrt pi / 2)"
proof -
let ?pI = "λf. (∫⇧+s. f (s::real) * indicator {0..} s ∂lborel)"
let ?gauss = "λx. exp (- x⇧2)"
let ?I = "indicator {0<..} :: real ⇒ real"
let ?ff= "λx s. x * exp (- x⇧2 * (1 + s⇧2)) :: real"
have *: "?pI ?gauss = (∫⇧+x. ?gauss x * ?I x ∂lborel)"
by (intro nn_integral_cong_AE AE_I[where N="{0}"]) (auto split: split_indicator)
have "?pI ?gauss * ?pI ?gauss = (∫⇧+x. ∫⇧+s. ?gauss x * ?gauss s * ?I s * ?I x ∂lborel ∂lborel)"
by (auto simp: nn_integral_cmult[symmetric] nn_integral_multc[symmetric] * ennreal_mult[symmetric]
intro!: nn_integral_cong split: split_indicator)
also have "… = (∫⇧+x. ∫⇧+s. ?ff x s * ?I s * ?I x ∂lborel ∂lborel)"
proof (rule nn_integral_cong, cases)
fix x :: real assume "x ≠ 0"
then show "(∫⇧+s. ?gauss x * ?gauss s * ?I s * ?I x ∂lborel) = (∫⇧+s. ?ff x s * ?I s * ?I x ∂lborel)"
by (subst nn_integral_real_affine[where t="0" and c="x"])
(auto simp: mult_exp_exp nn_integral_cmult[symmetric] field_simps zero_less_mult_iff ennreal_mult[symmetric]
intro!: nn_integral_cong split: split_indicator)
qed simp
also have "... = ∫⇧+s. ∫⇧+x. ?ff x s * ?I s * ?I x ∂lborel ∂lborel"
by (rule lborel_pair.Fubini'[symmetric]) auto
also have "... = ?pI (λs. ?pI (λx. ?ff x s))"
by (rule nn_integral_cong_AE)
(auto intro!: nn_integral_cong_AE AE_I[where N="{0}"] split: split_indicator_asm)
also have "… = ?pI (λs. ennreal (1 / (2 * (1 + s⇧2))))"
proof (intro nn_integral_cong ennreal_mult_right_cong)
fix s :: real show "?pI (λx. ?ff x s) = ennreal (1 / (2 * (1 + s⇧2)))"
proof (subst nn_integral_FTC_atLeast)
have "((λa. - (exp (- (a⇧2 * (1 + s⇧2))) / (2 + 2 * s⇧2))) ⤏ (- (0 / (2 + 2 * s⇧2)))) at_top"
apply (intro tendsto_intros filterlim_compose[OF exp_at_bot] filterlim_compose[OF filterlim_uminus_at_bot_at_top])
apply (subst mult.commute)
apply (auto intro!: filterlim_tendsto_pos_mult_at_top
filterlim_at_top_mult_at_top[OF filterlim_ident filterlim_ident]
simp: add_pos_nonneg power2_eq_square add_nonneg_eq_0_iff)
done
then show "((λa. - (exp (- a⇧2 - s⇧2 * a⇧2) / (2 + 2 * s⇧2))) ⤏ 0) at_top"
by (simp add: field_simps)
qed (auto intro!: derivative_eq_intros simp: field_simps add_nonneg_eq_0_iff)
qed
also have "... = ennreal (pi / 4)"
proof (subst nn_integral_FTC_atLeast)
show "((λa. arctan a / 2) ⤏ (pi / 2) / 2 ) at_top"
by (intro tendsto_intros) (simp_all add: tendsto_arctan_at_top)
qed (auto intro!: derivative_eq_intros simp: add_nonneg_eq_0_iff field_simps power2_eq_square)
finally have "?pI ?gauss^2 = pi / 4"
by (simp add: power2_eq_square)
then have "?pI ?gauss = sqrt (pi / 4)"
using power_eq_iff_eq_base[of 2 "enn2real (?pI ?gauss)" "sqrt (pi / 4)"]
by (cases "?pI ?gauss") (auto simp: power2_eq_square ennreal_mult[symmetric] ennreal_top_mult)
also have "?pI ?gauss = (∫⇧+x. indicator {0..} x *⇩R exp (- x⇧2) ∂lborel)"
by (intro nn_integral_cong) (simp split: split_indicator)
also have "sqrt (pi / 4) = sqrt pi / 2"
by (simp add: real_sqrt_divide)
finally show ?thesis
by (rule has_bochner_integral_nn_integral[rotated 3])
auto
qed
lemma gaussian_moment_1:
"has_bochner_integral lborel (λx::real. indicator {0..} x *⇩R (exp (- x⇧2) * x)) (1 / 2)"
proof -
have "(∫⇧+x. indicator {0..} x *⇩R (exp (- x⇧2) * x) ∂lborel) =
(∫⇧+x. ennreal (x * exp (- x⇧2)) * indicator {0..} x ∂lborel)"
by (intro nn_integral_cong)
(auto simp: ac_simps split: split_indicator)
also have "… = ennreal (0 - (- exp (- 0⇧2) / 2))"
proof (rule nn_integral_FTC_atLeast)
have "((λx::real. - exp (- x⇧2) / 2) ⤏ - 0 / 2) at_top"
by (intro tendsto_divide tendsto_minus filterlim_compose[OF exp_at_bot]
filterlim_compose[OF filterlim_uminus_at_bot_at_top]
filterlim_pow_at_top filterlim_ident)
auto
then show "((λa::real. - exp (- a⇧2) / 2) ⤏ 0) at_top"
by simp
qed (auto intro!: derivative_eq_intros)
also have "… = ennreal (1 / 2)"
by simp
finally show ?thesis
by (rule has_bochner_integral_nn_integral[rotated 3])
(auto split: split_indicator)
qed
lemma
fixes k :: nat
shows gaussian_moment_even_pos:
"has_bochner_integral lborel (λx::real. indicator {0..} x *⇩R (exp (-x⇧2)*x^(2 * k)))
((sqrt pi / 2) * (fact (2 * k) / (2 ^ (2 * k) * fact k)))"
(is "?even")
and gaussian_moment_odd_pos:
"has_bochner_integral lborel (λx::real. indicator {0..} x *⇩R (exp (-x⇧2)*x^(2 * k + 1))) (fact k / 2)"
(is "?odd")
proof -
let ?M = "λk x. exp (- x⇧2) * x^k :: real"
{ fix k I assume Mk: "has_bochner_integral lborel (λx. indicator {0..} x *⇩R ?M k x) I"
have "2 ≠ (0::real)"
by linarith
let ?f = "λb. ∫x. indicator {0..} x *⇩R ?M (k + 2) x * indicator {..b} x ∂lborel"
have "((λb. (k + 1) / 2 * (∫x. indicator {..b} x *⇩R (indicator {0..} x *⇩R ?M k x) ∂lborel) - ?M (k + 1) b / 2) ⤏
(k + 1) / 2 * (∫x. indicator {0..} x *⇩R ?M k x ∂lborel) - 0 / 2) at_top" (is ?tendsto)
proof (intro tendsto_intros ‹2 ≠ 0› tendsto_integral_at_top sets_lborel Mk[THEN integrable.intros])
show "(?M (k + 1) ⤏ 0) at_top"
proof cases
assume "even k"
have "((λx. ((x⇧2)^(k div 2 + 1) / exp (x⇧2)) * (1 / x) :: real) ⤏ 0 * 0) at_top"
by (intro tendsto_intros tendsto_divide_0[OF tendsto_const] filterlim_compose[OF tendsto_power_div_exp_0]
filterlim_at_top_imp_at_infinity filterlim_ident filterlim_pow_at_top filterlim_ident)
auto
also have "(λx. ((x⇧2)^(k div 2 + 1) / exp (x⇧2)) * (1 / x) :: real) = ?M (k + 1)"
using ‹even k› by (auto simp: fun_eq_iff exp_minus field_simps power2_eq_square power_mult elim: evenE)
finally show ?thesis by simp
next
assume "odd k"
have "((λx. ((x⇧2)^((k - 1) div 2 + 1) / exp (x⇧2)) :: real) ⤏ 0) at_top"
by (intro filterlim_compose[OF tendsto_power_div_exp_0] filterlim_at_top_imp_at_infinity
filterlim_ident filterlim_pow_at_top)
auto
also have "(λx. ((x⇧2)^((k - 1) div 2 + 1) / exp (x⇧2)) :: real) = ?M (k + 1)"
using ‹odd k› by (auto simp: fun_eq_iff exp_minus field_simps power2_eq_square power_mult elim: oddE)
finally show ?thesis by simp
qed
qed
also have "?tendsto ⟷ ((?f ⤏ (k + 1) / 2 * (∫x. indicator {0..} x *⇩R ?M k x ∂lborel) - 0 / 2) at_top)"
proof (intro filterlim_cong refl eventually_at_top_linorder[THEN iffD2] exI[of _ 0] allI impI)
fix b :: real assume b: "0 ≤ b"
have "Suc k * (∫x. indicator {0..b} x *⇩R ?M k x ∂lborel) = (∫x. indicator {0..b} x *⇩R (exp (- x⇧2) * ((Suc k) * x ^ k)) ∂lborel)"
unfolding integral_mult_right_zero[symmetric] by (intro integral_cong) auto
also have "… = exp (- b⇧2) * b ^ (Suc k) - exp (- 0⇧2) * 0 ^ (Suc k) -
(∫x. indicator {0..b} x *⇩R (- 2 * x * exp (- x⇧2) * x ^ (Suc k)) ∂lborel)"
by (rule integral_by_parts')
(auto intro!: derivative_eq_intros b
simp: diff_Suc of_nat_Suc field_simps split: nat.split)
also have "(∫x. indicator {0..b} x *⇩R (- 2 * x * exp (- x⇧2) * x ^ (Suc k)) ∂lborel) =
(∫x. indicator {0..b} x *⇩R (- 2 * (exp (- x⇧2) * x ^ (k + 2))) ∂lborel)"
by (intro integral_cong) auto
finally have "Suc k * (∫x. indicator {0..b} x *⇩R ?M k x ∂lborel) =
exp (- b⇧2) * b ^ (Suc k) + 2 * (∫x. indicator {0..b} x *⇩R ?M (k + 2) x ∂lborel)"
by (simp del: real_scaleR_def integral_mult_right add: integral_mult_right[symmetric])
then show "(k + 1) / 2 * (∫x. indicator {..b} x *⇩R (indicator {0..} x *⇩R ?M k x)∂lborel) - exp (- b⇧2) * b ^ (k + 1) / 2 = ?f b"
by (simp add: field_simps atLeastAtMost_def indicator_inter_arith)
qed
finally have int_M_at_top: "((?f ⤏ (k + 1) / 2 * (∫x. indicator {0..} x *⇩R ?M k x ∂lborel)) at_top)"
by simp
have "has_bochner_integral lborel (λx. indicator {0..} x *⇩R ?M (k + 2) x) ((k + 1) / 2 * I)"
proof (rule has_bochner_integral_monotone_convergence_at_top)
fix y :: real
have *: "(λx. indicator {0..} x *⇩R ?M (k + 2) x * indicator {..y} x::real) =
(λx. indicator {0..y} x *⇩R ?M (k + 2) x)"
by rule (simp split: split_indicator)
show "integrable lborel (λx. indicator {0..} x *⇩R (?M (k + 2) x) * indicator {..y} x::real)"
unfolding * by (rule borel_integrable_compact) (auto intro!: continuous_intros)
show "((?f ⤏ (k + 1) / 2 * I) at_top)"
using int_M_at_top has_bochner_integral_integral_eq[OF Mk] by simp
qed (auto split: split_indicator) }
note step = this
show ?even
proof (induct k)
case (Suc k)
note step[OF this]
also have "(real (2 * k + 1) / 2 * (sqrt pi / 2 * ((fact (2 * k)) / ((2::real)^(2*k) * fact k)))) =
sqrt pi / 2 * ((fact (2 * Suc k)) / ((2::real)^(2 * Suc k) * fact (Suc k)))"
apply (simp add: field_simps del: fact_Suc)
apply (simp add: of_nat_mult field_simps)
done
finally show ?case
by simp
qed (insert gaussian_moment_0, simp)
show ?odd
proof (induct k)
case (Suc k)
note step[OF this]
also have "(real (2 * k + 1 + 1) / (2::real) * ((fact k) / 2)) = (fact (Suc k)) / 2"
by (simp add: field_simps of_nat_Suc divide_simps del: fact_Suc) (simp add: field_simps)
finally show ?case
by simp
qed (insert gaussian_moment_1, simp)
qed
context
fixes k :: nat and μ σ :: real assumes [arith]: "0 < σ"
begin
lemma normal_moment_even:
"has_bochner_integral lborel (λx. normal_density μ σ x * (x - μ) ^ (2 * k)) (fact (2 * k) / ((2 / σ⇧2)^k * fact k))"
proof -
have eq: "⋀x::real. x⇧2^k = (x^k)⇧2"
by (simp add: power_mult[symmetric] ac_simps)
have "has_bochner_integral lborel (λx. exp (-x⇧2)*x^(2 * k))
(sqrt pi * (fact (2 * k) / (2 ^ (2 * k) * fact k)))"
using has_bochner_integral_even_function[OF gaussian_moment_even_pos[where k=k]] by simp
then have "has_bochner_integral lborel (λx. (exp (-x⇧2)*x^(2 * k)) * ((2*σ⇧2)^k / sqrt (2 * pi * σ⇧2)))
((sqrt pi * (fact (2 * k) / (2 ^ (2 * k) * fact k))) * ((2*σ⇧2)^k / sqrt (2 * pi * σ⇧2)))"
by (rule has_bochner_integral_mult_left)
also have "(λx. (exp (-x⇧2)*x^(2 * k)) * ((2*σ⇧2)^k / sqrt (2 * pi * σ⇧2))) =
(λx. exp (- ((sqrt 2 * σ) * x)⇧2 / (2*σ⇧2)) * ((sqrt 2 * σ) * x) ^ (2 * k) / sqrt (2 * pi * σ⇧2))"
by (auto simp: fun_eq_iff field_simps real_sqrt_power[symmetric] real_sqrt_mult
real_sqrt_divide power_mult eq)
also have "((sqrt pi * (fact (2 * k) / (2 ^ (2 * k) * fact k))) * ((2*σ⇧2)^k / sqrt (2 * pi * σ⇧2))) =
(inverse (sqrt 2 * σ) * ((fact (2 * k))) / ((2/σ⇧2) ^ k * (fact k)))"
by (auto simp: fun_eq_iff power_mult field_simps real_sqrt_power[symmetric] real_sqrt_mult
power2_eq_square)
finally show ?thesis
unfolding normal_density_def
by (subst lborel_has_bochner_integral_real_affine_iff[where c="sqrt 2 * σ" and t=μ]) simp_all
qed
lemma normal_moment_abs_odd:
"has_bochner_integral lborel (λx. normal_density μ σ x * ¦x - μ¦^(2 * k + 1)) (2^k * σ^(2 * k + 1) * fact k * sqrt (2 / pi))"
proof -
have "has_bochner_integral lborel (λx::real. indicator {0..} x *⇩R (exp (-x⇧2)*¦x¦^(2 * k + 1))) (fact k / 2)"
by (rule has_bochner_integral_cong[THEN iffD1, OF _ _ _ gaussian_moment_odd_pos[of k]]) auto
from has_bochner_integral_even_function[OF this]
have "has_bochner_integral lborel (λx::real. exp (-x⇧2)*¦x¦^(2 * k + 1)) (fact k)"
by simp
then have "has_bochner_integral lborel (λx. (exp (-x⇧2)*¦x¦^(2 * k + 1)) * (2^k * σ^(2 * k + 1) / sqrt (pi * σ⇧2)))
(fact k * (2^k * σ^(2 * k + 1) / sqrt (pi * σ⇧2)))"
by (rule has_bochner_integral_mult_left)
also have "(λx. (exp (-x⇧2)*¦x¦^(2 * k + 1)) * (2^k * σ^(2 * k + 1) / sqrt (pi * σ⇧2))) =
(λx. exp (- (((sqrt 2 * σ) * x)⇧2 / (2 * σ⇧2))) * ¦sqrt 2 * σ * x¦ ^ (2 * k + 1) / sqrt (2 * pi * σ⇧2))"
by (simp add: field_simps abs_mult real_sqrt_power[symmetric] power_mult real_sqrt_mult)
also have "(fact k * (2^k * σ^(2 * k + 1) / sqrt (pi * σ⇧2))) =
(inverse (sqrt 2) * inverse σ * (2 ^ k * (σ * σ ^ (2 * k)) * (fact k) * sqrt (2 / pi)))"
by (auto simp: fun_eq_iff power_mult field_simps real_sqrt_power[symmetric] real_sqrt_divide
real_sqrt_mult)
finally show ?thesis
unfolding normal_density_def
by (subst lborel_has_bochner_integral_real_affine_iff[where c="sqrt 2 * σ" and t=μ])
simp_all
qed
lemma normal_moment_odd:
"has_bochner_integral lborel (λx. normal_density μ σ x * (x - μ)^(2 * k + 1)) 0"
proof -
have "has_bochner_integral lborel (λx. exp (- x⇧2) * x^(2 * k + 1)::real) 0"
using gaussian_moment_odd_pos by (rule has_bochner_integral_odd_function) simp
then have "has_bochner_integral lborel (λx. (exp (-x⇧2)*x^(2 * k + 1)) * (2^k*σ^(2*k)/sqrt pi))
(0 * (2^k*σ^(2*k)/sqrt pi))"
by (rule has_bochner_integral_mult_left)
also have "(λx. (exp (-x⇧2)*x^(2 * k + 1)) * (2^k*σ^(2*k)/sqrt pi)) =
(λx. exp (- ((sqrt 2 * σ * x)⇧2 / (2 * σ⇧2))) *
(sqrt 2 * σ * x * (sqrt 2 * σ * x) ^ (2 * k)) /
sqrt (2 * pi * σ⇧2))"
unfolding real_sqrt_mult
by (simp add: field_simps abs_mult real_sqrt_power[symmetric] power_mult fun_eq_iff)
finally show ?thesis
unfolding normal_density_def
by (subst lborel_has_bochner_integral_real_affine_iff[where c="sqrt 2 * σ" and t=μ]) simp_all
qed
lemma integral_normal_moment_even:
"integral⇧L lborel (λx. normal_density μ σ x * (x - μ)^(2 * k)) = fact (2 * k) / ((2 / σ⇧2)^k * fact k)"
using normal_moment_even by (rule has_bochner_integral_integral_eq)
lemma integral_normal_moment_abs_odd:
"integral⇧L lborel (λx. normal_density μ σ x * ¦x - μ¦^(2 * k + 1)) = 2 ^ k * σ ^ (2 * k + 1) * fact k * sqrt (2 / pi)"
using normal_moment_abs_odd by (rule has_bochner_integral_integral_eq)
lemma integral_normal_moment_odd:
"integral⇧L lborel (λx. normal_density μ σ x * (x - μ)^(2 * k + 1)) = 0"
using normal_moment_odd by (rule has_bochner_integral_integral_eq)
end
context
fixes σ :: real
assumes σ_pos[arith]: "0 < σ"
begin
lemma normal_moment_nz_1: "has_bochner_integral lborel (λx. normal_density μ σ x * x) μ"
proof -
note normal_moment_even[OF σ_pos, of μ 0]
note normal_moment_odd[OF σ_pos, of μ 0] has_bochner_integral_mult_left[of μ, OF this]
note has_bochner_integral_add[OF this]
then show ?thesis
by (simp add: power2_eq_square field_simps)
qed
lemma integral_normal_moment_nz_1:
"integral⇧L lborel (λx. normal_density μ σ x * x) = μ"
using normal_moment_nz_1 by (rule has_bochner_integral_integral_eq)
lemma integrable_normal_moment_nz_1: "integrable lborel (λx. normal_density μ σ x * x)"
using normal_moment_nz_1 by rule
lemma integrable_normal_moment: "integrable lborel (λx. normal_density μ σ x * (x - μ)^k)"
proof cases
assume "even k" then show ?thesis
using integrable.intros[OF normal_moment_even] by (auto elim: evenE)
next
assume "odd k" then show ?thesis
using integrable.intros[OF normal_moment_odd] by (auto elim: oddE)
qed
lemma integrable_normal_moment_abs: "integrable lborel (λx. normal_density μ σ x * ¦x - μ¦^k)"
proof cases
assume "even k" then show ?thesis
using integrable.intros[OF normal_moment_even] by (auto simp add: power_even_abs elim: evenE)
next
assume "odd k" then show ?thesis
using integrable.intros[OF normal_moment_abs_odd] by (auto elim: oddE)
qed
lemma integrable_normal_density[simp, intro]: "integrable lborel (normal_density μ σ)"
using integrable_normal_moment[of μ 0] by simp
lemma integral_normal_density[simp]: "(∫x. normal_density μ σ x ∂lborel) = 1"
using integral_normal_moment_even[of σ μ 0] by simp
lemma prob_space_normal_density:
"prob_space (density lborel (normal_density μ σ))"
proof qed (simp add: emeasure_density nn_integral_eq_integral normal_density_nonneg)
end
context
fixes k :: nat
begin
lemma std_normal_moment_even:
"has_bochner_integral lborel (λx. std_normal_density x * x ^ (2 * k)) (fact (2 * k) / (2^k * fact k))"
using normal_moment_even[of 1 0 k] by simp
lemma std_normal_moment_abs_odd:
"has_bochner_integral lborel (λx. std_normal_density x * ¦x¦^(2 * k + 1)) (sqrt (2/pi) * 2^k * fact k)"
using normal_moment_abs_odd[of 1 0 k] by (simp add: ac_simps)
lemma std_normal_moment_odd:
"has_bochner_integral lborel (λx. std_normal_density x * x^(2 * k + 1)) 0"
using normal_moment_odd[of 1 0 k] by simp
lemma integral_std_normal_moment_even:
"integral⇧L lborel (λx. std_normal_density x * x^(2*k)) = fact (2 * k) / (2^k * fact k)"
using std_normal_moment_even by (rule has_bochner_integral_integral_eq)
lemma integral_std_normal_moment_abs_odd:
"integral⇧L lborel (λx. std_normal_density x * ¦x¦^(2 * k + 1)) = sqrt (2 / pi) * 2^k * fact k"
using std_normal_moment_abs_odd by (rule has_bochner_integral_integral_eq)
lemma integral_std_normal_moment_odd:
"integral⇧L lborel (λx. std_normal_density x * x^(2 * k + 1)) = 0"
using std_normal_moment_odd by (rule has_bochner_integral_integral_eq)
lemma integrable_std_normal_moment_abs: "integrable lborel (λx. std_normal_density x * ¦x¦^k)"
using integrable_normal_moment_abs[of 1 0 k] by simp
lemma integrable_std_normal_moment: "integrable lborel (λx. std_normal_density x * x^k)"
using integrable_normal_moment[of 1 0 k] by simp
end
lemma (in prob_space) normal_density_affine:
assumes X: "distributed M lborel X (normal_density μ σ)"
assumes [simp, arith]: "0 < σ" "α ≠ 0"
shows "distributed M lborel (λx. β + α * X x) (normal_density (β + α * μ) (¦α¦ * σ))"
proof -
have eq: "⋀x. ¦α¦ * normal_density (β + α * μ) (¦α¦ * σ) (x * α + β) =
normal_density μ σ x"
by (simp add: normal_density_def real_sqrt_mult field_simps)
(simp add: power2_eq_square field_simps)
show ?thesis
by (rule distributed_affineI[OF _ ‹α ≠ 0›, where t=β])
(simp_all add: eq X ennreal_mult'[symmetric])
qed
lemma (in prob_space) normal_standard_normal_convert:
assumes pos_var[simp, arith]: "0 < σ"
shows "distributed M lborel X (normal_density μ σ) = distributed M lborel (λx. (X x - μ) / σ) std_normal_density"
proof auto
assume "distributed M lborel X (λx. ennreal (normal_density μ σ x))"
then have "distributed M lborel (λx. -μ / σ + (1/σ) * X x) (λx. ennreal (normal_density (-μ / σ + (1/σ)* μ) (¦1/σ¦ * σ) x))"
by(rule normal_density_affine) auto
then show "distributed M lborel (λx. (X x - μ) / σ) (λx. ennreal (std_normal_density x))"
by (simp add: diff_divide_distrib[symmetric] field_simps)
next
assume *: "distributed M lborel (λx. (X x - μ) / σ) (λx. ennreal (std_normal_density x))"
have "distributed M lborel (λx. μ + σ * ((X x - μ) / σ)) (λx. ennreal (normal_density μ ¦σ¦ x))"
using normal_density_affine[OF *, of σ μ] by simp
then show "distributed M lborel X (λx. ennreal (normal_density μ σ x))" by simp
qed
lemma conv_normal_density_zero_mean:
assumes [simp, arith]: "0 < σ" "0 < τ"
shows "(λx. ∫⇧+y. ennreal (normal_density 0 σ (x - y) * normal_density 0 τ y) ∂lborel) =
normal_density 0 (sqrt (σ⇧2 + τ⇧2))" (is "?LHS = ?RHS")
proof -
def σ' ≡ "σ⇧2" and τ' ≡ "τ⇧2"
then have [simp, arith]: "0 < σ'" "0 < τ'"
by simp_all
let ?σ = "sqrt ((σ' * τ') / (σ' + τ'))"
have sqrt: "(sqrt (2 * pi * (σ' + τ')) * sqrt (2 * pi * (σ' * τ') / (σ' + τ'))) =
(sqrt (2 * pi * σ') * sqrt (2 * pi * τ'))"
by (subst power_eq_iff_eq_base[symmetric, where n=2])
(simp_all add: real_sqrt_mult[symmetric] power2_eq_square)
have "?LHS =
(λx. ∫⇧+y. ennreal((normal_density 0 (sqrt (σ' + τ')) x) * normal_density (τ' * x / (σ' + τ')) ?σ y) ∂lborel)"
apply (intro ext nn_integral_cong)
apply (simp add: normal_density_def σ'_def[symmetric] τ'_def[symmetric] sqrt mult_exp_exp)
apply (simp add: divide_simps power2_eq_square)
apply (simp add: field_simps)
done
also have "... =
(λx. (normal_density 0 (sqrt (σ⇧2 + τ⇧2)) x) * ∫⇧+y. ennreal( normal_density (τ⇧2* x / (σ⇧2 + τ⇧2)) ?σ y) ∂lborel)"
by (subst nn_integral_cmult[symmetric])
(auto simp: σ'_def τ'_def normal_density_def ennreal_mult'[symmetric])
also have "... = (λx. (normal_density 0 (sqrt (σ⇧2 + τ⇧2)) x))"
by (subst nn_integral_eq_integral) (auto simp: normal_density_nonneg)
finally show ?thesis by fast
qed
lemma conv_std_normal_density:
"(λx. ∫⇧+y. ennreal (std_normal_density (x - y) * std_normal_density y) ∂lborel) =
(normal_density 0 (sqrt 2))"
by (subst conv_normal_density_zero_mean) simp_all
lemma (in prob_space) sum_indep_normal:
assumes indep: "indep_var borel X borel Y"
assumes pos_var[arith]: "0 < σ" "0 < τ"
assumes normalX[simp]: "distributed M lborel X (normal_density μ σ)"
assumes normalY[simp]: "distributed M lborel Y (normal_density ν τ)"
shows "distributed M lborel (λx. X x + Y x) (normal_density (μ + ν) (sqrt (σ⇧2 + τ⇧2)))"
proof -
have ind[simp]: "indep_var borel (λx. - μ + X x) borel (λx. - ν + Y x)"
proof -
have "indep_var borel ( (λx. -μ + x) o X) borel ((λx. - ν + x) o Y)"
by (auto intro!: indep_var_compose assms)
then show ?thesis by (simp add: o_def)
qed
have "distributed M lborel (λx. -μ + 1 * X x) (normal_density (- μ + 1 * μ) (¦1¦ * σ))"
by(rule normal_density_affine[OF normalX pos_var(1), of 1 "-μ"]) simp
then have 1[simp]: "distributed M lborel (λx. - μ + X x) (normal_density 0 σ)" by simp
have "distributed M lborel (λx. -ν + 1 * Y x) (normal_density (- ν + 1 * ν) (¦1¦ * τ))"
by(rule normal_density_affine[OF normalY pos_var(2), of 1 "-ν"]) simp
then have 2[simp]: "distributed M lborel (λx. - ν + Y x) (normal_density 0 τ)" by simp
have *: "distributed M lborel (λx. (- μ + X x) + (- ν + Y x)) (λx. ennreal (normal_density 0 (sqrt (σ⇧2 + τ⇧2)) x))"
using distributed_convolution[OF ind 1 2] conv_normal_density_zero_mean[OF pos_var]
by (simp add: ennreal_mult'[symmetric] normal_density_nonneg)
have "distributed M lborel (λx. μ + ν + 1 * (- μ + X x + (- ν + Y x)))
(λx. ennreal (normal_density (μ + ν + 1 * 0) (¦1¦ * sqrt (σ⇧2 + τ⇧2)) x))"
by (rule normal_density_affine[OF *, of 1 "μ + ν"]) (auto simp: add_pos_pos)
then show ?thesis by auto
qed
lemma (in prob_space) diff_indep_normal:
assumes indep[simp]: "indep_var borel X borel Y"
assumes [simp, arith]: "0 < σ" "0 < τ"
assumes normalX[simp]: "distributed M lborel X (normal_density μ σ)"
assumes normalY[simp]: "distributed M lborel Y (normal_density ν τ)"
shows "distributed M lborel (λx. X x - Y x) (normal_density (μ - ν) (sqrt (σ⇧2 + τ⇧2)))"
proof -
have "distributed M lborel (λx. 0 + - 1 * Y x) (λx. ennreal (normal_density (0 + - 1 * ν) (¦- 1¦ * τ) x))"
by(rule normal_density_affine, auto)
then have [simp]:"distributed M lborel (λx. - Y x) (λx. ennreal (normal_density (- ν) τ x))" by simp
have "distributed M lborel (λx. X x + (- Y x)) (normal_density (μ + - ν) (sqrt (σ⇧2 + τ⇧2)))"
apply (rule sum_indep_normal)
apply (rule indep_var_compose[unfolded comp_def, of borel _ borel _ "λx. x" _ "λx. - x"])
apply simp_all
done
then show ?thesis by simp
qed
lemma (in prob_space) setsum_indep_normal:
assumes "finite I" "I ≠ {}" "indep_vars (λi. borel) X I"
assumes "⋀i. i ∈ I ⟹ 0 < σ i"
assumes normal: "⋀i. i ∈ I ⟹ distributed M lborel (X i) (normal_density (μ i) (σ i))"
shows "distributed M lborel (λx. ∑i∈I. X i x) (normal_density (∑i∈I. μ i) (sqrt (∑i∈I. (σ i)⇧2)))"
using assms
proof (induct I rule: finite_ne_induct)
case (singleton i) then show ?case by (simp add : abs_of_pos)
next
case (insert i I)
then have 1: "distributed M lborel (λx. (X i x) + (∑i∈I. X i x))
(normal_density (μ i + setsum μ I) (sqrt ((σ i)⇧2 + (sqrt (∑i∈I. (σ i)⇧2))⇧2)))"
apply (intro sum_indep_normal indep_vars_setsum insert real_sqrt_gt_zero)
apply (auto intro: indep_vars_subset intro!: setsum_pos)
apply fastforce
done
have 2: "(λx. (X i x)+ (∑i∈I. X i x)) = (λx. (∑j∈insert i I. X j x))"
"μ i + setsum μ I = setsum μ (insert i I)"
using insert by auto
have 3: "(sqrt ((σ i)⇧2 + (sqrt (∑i∈I. (σ i)⇧2))⇧2)) = (sqrt (∑i∈insert i I. (σ i)⇧2))"
using insert by (simp add: setsum_nonneg)
show ?case using 1 2 3 by simp
qed
lemma (in prob_space) standard_normal_distributed_expectation:
assumes D: "distributed M lborel X std_normal_density"
shows "expectation X = 0"
using integral_std_normal_moment_odd[of 0]
distributed_integral[OF D, of "λx. x", symmetric]
by (auto simp: )
lemma (in prob_space) normal_distributed_expectation:
assumes σ[arith]: "0 < σ"
assumes D: "distributed M lborel X (normal_density μ σ)"
shows "expectation X = μ"
using integral_normal_moment_nz_1[OF σ, of μ] distributed_integral[OF D, of "λx. x", symmetric]
by (auto simp: field_simps)
lemma (in prob_space) normal_distributed_variance:
fixes a b :: real
assumes [simp, arith]: "0 < σ"
assumes D: "distributed M lborel X (normal_density μ σ)"
shows "variance X = σ⇧2"
using integral_normal_moment_even[of σ μ 1]
by (subst distributed_integral[OF D, symmetric])
(simp_all add: eval_nat_numeral normal_distributed_expectation[OF assms])
lemma (in prob_space) standard_normal_distributed_variance:
"distributed M lborel X std_normal_density ⟹ variance X = 1"
using normal_distributed_variance[of 1 X 0] by simp
lemma (in information_space) entropy_normal_density:
assumes [arith]: "0 < σ"
assumes D: "distributed M lborel X (normal_density μ σ)"
shows "entropy b lborel X = log b (2 * pi * exp 1 * σ⇧2) / 2"
proof -
have "entropy b lborel X = - (∫ x. normal_density μ σ x * log b (normal_density μ σ x) ∂lborel)"
using D by (rule entropy_distr) simp
also have "… = - (∫ x. normal_density μ σ x * (- ln (2 * pi * σ⇧2) - (x - μ)⇧2 / σ⇧2) / (2 * ln b) ∂lborel)"
by (intro arg_cong[where f="uminus"] integral_cong)
(auto simp: normal_density_def field_simps ln_mult log_def ln_div ln_sqrt)
also have "… = - (∫x. - (normal_density μ σ x * (ln (2 * pi * σ⇧2)) + (normal_density μ σ x * (x - μ)⇧2) / σ⇧2) / (2 * ln b) ∂lborel)"
by (intro arg_cong[where f="uminus"] integral_cong) (auto simp: divide_simps field_simps)
also have "… = (∫x. normal_density μ σ x * (ln (2 * pi * σ⇧2)) + (normal_density μ σ x * (x - μ)⇧2) / σ⇧2 ∂lborel) / (2 * ln b)"
by (simp del: minus_add_distrib)
also have "… = (ln (2 * pi * σ⇧2) + 1) / (2 * ln b)"
using integral_normal_moment_even[of σ μ 1] by (simp add: integrable_normal_moment fact_numeral)
also have "… = log b (2 * pi * exp 1 * σ⇧2) / 2"
by (simp add: log_def field_simps ln_mult)
finally show ?thesis .
qed
end