), P{|Zn – c| > ɛ} → 0 as n → o. Show that for any b on g, E[g(Z„)] → g(c) as n→o
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Question in image
![8.4. Let Zn,n 2 1, be a sequence of random variables and ca constant such
that for each ɛ > 0, P{|Z, – c| > ɛ}→ 0 as n → o. Show that for any bounded
continuous function g,
E[g(Zn)] → g(c) as
n → 00](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa438126-e53d-43ea-992d-aeb965a47556%2F9a39656b-68be-4fcf-b0ea-8245764e242e%2F1r9uplb_processed.png&w=3840&q=75)
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- Let X1,X2,... be a sequence of identically distributed random variables with E|X1|<∞ and let Yn = n−1max1≤i≤n|Xi|. Show that limnE(Yn) = 0Suppose X1, X2, ... , Xn is a random sample and Xi = {1, with probability p 0, with probability 1-p} for every i = 1, 2, ... , n. Find the Moment Generating Function of ∑i=1n Xi . What is the distribution of ∑i=1n Xi ?Let X1, X2, ... , Xn be a random sample from N(μ, σ2). Find the Moment Generating Function of X̅. If n = 16 and σ = 2, compute P(-1 ≤ X̅ - μ ≤ 1).
- Let X1, X2, X3, . . . be a sequence of independent Poisson distributed random variables with parameter 1. For n ≥ 1 let Sn = X1 + · · · + Xn. (a) Show that GXi(s) = es−1.(b) Deduce from part (a) that GSn(s) = ens−n.Let U₁,..., Un be i.i.d. random variables uniformly distributed in [0, 1] and let Mn = max Ui . 1<i<n Find the cdf of Mn, which we denote by G (t), for t = [0, 1]. For t = [0, 1], G (t) = Now, let Fn (t) denote the cdf of n (1 - Mn); for t > 0, compute lim F₁ (t) =Let Xi be IID random variables which have the same law as X. Let L(t) = E(e^tX.) Suppose that this is well defined for t ∈ [−1, 1]. Express the moment generating function of the Sum from i=1 to k Xi in terms of k and L
- Let X1,…,Xn be i.i.d. uniform random variables in [0,θ], for some θ>0. Denote by Mn=maxi=1,…,nXi. Compute the cumulative distribution function Fn(t) of n(1−Mn/θ) for fixed t∈[0,n] and any positive integer n. Fn(t)= Compute the following limit. limn→∞Fn(t)= Next, we will use the previous question to find an interval I of the form I=[Mn,Mn+c], that does not depend on θ and such that P[I∋θ]→.95, as n→∞. The strategy now is to use a plug-in estimator for θ to replace it in the expression for c. Parts (a) and (b) suggest that we use c of the form (tn)Mn, where t ought to equal a certain value in order for P[I∋θ]→.95. What is the appropriate numerical value of t? t=Let U1, U2, ... be i.i.d. Uni(0, 1) random variables. - log U; is distributed as exp(1). (2) Find the limit of the random sequence (U1U2 · · Un)"", as n → o. (3) Describe what happens to the sequence of random variables eVn (U¡U2 · · · Un)'/v™, (1) Show that X; as n → ∞?Let Z1,...,Zn be a sequence of random variables. For every n = 1, 2;..., P(Zn =n^2) = 1/n, and P(Zn = 0) = 1 - 1/n. In other words, Zn is binary. Show that(a) limit as n goes to infinity E(Zn)= infinity
- 4.3.2 Let Y ∼ Uniform[0, 1], and let Xn = Yn. Prove that Xn → 0 with probability 1.7. Assume that X₁,..., Xn is a random sample from a Bernoulli (p) and let Yn n 1/1 X₁. For p # 1/2, Determine the asymptotic distribution of √n[Yn(1 – Yn) - p(1 - p)]. 7. Assume that \( X_1, ..., X_n \) is a random sample from a Bernoulli(p) and let \( {Y}_n = \1/n\sum_{i=1}^{n} X_i \). For \( p =/ 1/2), Determine the asymptotic distribution of \( \sqrt{n}({Y}_n(1 - {Y}_n) - p(1 - p)) \).Let X1, . . . , Xn, . . . ∼ iid Bern(θ). Consider the Bayes estimator under squared error loss with the Unif(0,1) prior. Show that this estimator is consistent.
