Let X> 0 and X, X₁, X2,... be random variables with X~ Poisson(A) and X₁ ~ Binom(n,A). Prove that {Xn}n>1 converges to X in distribution. You can assume that n is large enough such that < 1. n
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- The conditional probability of E given that F occurs is P(EF)=___________. So in rolling a die the conditional probability of the event E, getting a six, given that the event F, getting an even number, has occurred is P(EF)=___________.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 Xi be arandom sample from U(0,1)prove that Xn’ convarges in probability to 0.50
- 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) = 0Let X be a random variable with pdf given by fX(x) = 1/[π(1 + x2)] for all real number x. Prove that X and 1/X are identically distributed by showing that they have the same probability distribution.Let Z be a N(0, 1) random variable, and let F(x) be thecumulative distribution function for Z. Show that on S (∞, 0], F(x) is an increasing convex function, and on S [0, ∞), F(x) is an increasing concave function.
- (b) Let Z be a discrete random variable with E(Z) = 0. Does it necessarily follow that E(Z³) = 0? If yes, give a proof; if no, give a counterexample.Let X be an exponential random variable with standard deviation σ. FindP(|X − E(X)| > kσ ) for k = 2, 3, 4, and compare the results to the boundsfrom Chebyshev’s inequality.We require a non-zero variance in the central limit theorem, because Select one: a. otherwise, the Xi do not converge in distribution. b. otherwise we cannot use the weak law of large numbers and the central limit theorem gives a stronger conclusion. c. otherwise the Xi are constant and therefore do not converge to the normal distribution. d. otherwise the Xi are all equal to the constant random variable 0.
- Let Q be a continuous random variable with PDFfQ(q)= 6q(1 − q) if 0 ≤ q ≤ 1fQ(q) = 0 otherwiseThis Q represents the probability of success of a Bernoulli random variable X, i.e.,P (X = 1 | Q = q) = q.Find fQ|X (q|x) for x ∈ {0, 1} and all q.If X is a continuous random variable with X ∼ Uniform([0, 2]), what is E[X^3]?If X1, X2, ... , Xn are independent random variables having identical Bernoulli distributions with the param-eter θ, then X is the proportion of successes in n trials, which we denote by ˆ . Verify that(a) E()ˆ = θ;(b) var()ˆ = θ (1 − θ )n .