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By Van Der Merwe A. J., Du Plessis J. L.

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For general f ∈ Lp we may choose some simple measurable functions fn → f with |fn | ≤ |f |. Since |fn −f |p ≤ 2p+1 |f |p , we get fn −f p → 0 by dominated convergence. ✷ Taking p = q = 2 and r = 1 in H¨older’s inequality (8), we get the CauchyBuniakovsky inequality (often called Schwarz’s inequality) fg 1 ≤ f 2 g 2. In particular, we note that, for any f, g ∈ L2 , the inner product f, g = µ(f g) exists and satisfies | f, g | ≤ f 2 g 2 . From the obvious bilinearity of the inner product, we get the parallelogram identity f +g 2 + f −g 2 =2 f 2 + 2 g 2, f, g ∈ L2 .

Then ξ = η iff F = G. 1. as ✷ The expected value, expectation, or mean of a random variable ξ is defined Eξ = Ω ξ dP = R x(P ◦ ξ −1 )(dx) (4) whenever either integral exists. 22. By the same result we note that, for any random element ξ in some measurable space S and for an arbitrary measurable function f : S → R, Ef (ξ) = = Ω R f (ξ) dP = S f (s)(P ◦ ξ −1 )(ds) x(P ◦ (f ◦ ξ)−1 )(dx), (5) 26 Foundations of Modern Probability provided that at least one of the three integrals exists. Integrals over a measurable subset A ⊂ Ω are often denoted by E[ξ; A] = E(ξ 1A ) = A ξ dP, A ∈ A.

10. Let ξ1 , ξ2 , . . d. random elements with distribution µ in some measurable space (S, S). Fix a set A ∈ S with µA > 0, and put τ = inf{k; ξk ∈ A}. Show that ξτ has distribution µ[ · |A] = µ(· ∩ A)/µA. 11. Let ξ1 , ξ2 , . . be independent random variables taking values in [0, 1]. Show that E n ξn = n Eξn . In particular, show that P n An = n P An for any independent events A1 , A2 , . . 12. Let ξ1 , ξ2 , . . be arbitrary random variables. Show that there exist some constants c1 , c2 , .

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A Bayesian Approach to Selection and Ranking Procedures: The Unequal Variance Case by Van Der Merwe A. J., Du Plessis J. L.


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