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20 Dec 2023: Set. A = {ω : ω An infinitely often}, B = {ω : ω An for all sufficiently large n}. ... Calculate the probability thatm given people will all be on the committee (a) directly, (b) using the inclusion-exclusion formula.Deduce that (. n mr m. )=. mj=0.

20 Dec 2023: a) Show that, for all p (0, ) and all x (0, ),. P(|X| x) E(|X|p)xp. (b) Show that, for all β 0,P(X x) E(eβX)eβx. ... 3. Let X be a Poisson random variable of parameter λ (0, ). (a) By optimizing the estimate of Question 2(b) over β, show that, for

31 Oct 2023: b) Let X Bin(n,θ), where θ Θ = [0, 1]. Find all prior distributions π on Θ for whichthe maximum likelihood estimator is a Bayes rule for estimating θ in ... a) Show that the James-Stein estimator θ̃(1) dominates all estimators θ̃(c),c 6= 1.(b)

3 Jun 2024: In all the questions that follow, X is an n by p design matrix with full column rank and H is theorthogonal projection on to the column space of X. ... Show that for all x Rn, ‖x‖2 ‖ΠWx‖2 ‖ΠV x‖2. (b) Consider the normal linear model with a

3 Jun 2024: f(y; µ,σ2) =1. 2πσ2e. (yµ)2. 2σ2 ,y R. (b) The Gamma distribution, Gamma(α,β):. ... 2 = 2/σ2βj for all j). (c) How do the expressions in (a) and (b) simplify when g(µi) is the canonical link function?

3 Jun 2024: j, j) B for all j V) or an directed acyclic graph (DAG) G = (V,D). ... Show that there exists a trek from j to k via U(meaning all its non-endpoints are in U) if and only if there exists an arc from j to

20 Feb 2024: 3. Let F be the set of all polynomials of degree at most 2 on X = Rp. ... Then g : Rm R. given by g(x) = f(Ax b) is a convex function.(c) Let Cα Rd be convex for all α I where I is some index set.

17 Jan 2024: xπ)>(z π) 0 for all z C. 4. Show that ‖β‖1 = {b : for each j,bj [1, 1] and bj = sgn(βj) if βj 6= 0}. ... c) Let. H =. {Mm=1. βmhm : ‖β‖1 1, hm B for m = 1,.

3 Jun 2024: adjustment set for A, B given S′ in G. Then every element in the output ofConfounderSelect(X, Y ) is a sufficient adjustment set for (X, Y ). 2 Completeness (all minimal primary ... all minimal sufficiency): Suppose further thatFindPrimary((A, B); S′)

3 Jun 2024: adjustment set for A, B given S′ in G. Then every element in the output ofConfounderSelect(X, Y ) is a sufficient adjustment set for (X, Y ). 2 Completeness (all minimal primary ... all minimal sufficiency): Suppose further thatFindPrimary((A, B); S′)