In the following problem, please select the correct answer.

Question

Let X be a non-negative random variable. Then, for any a>0, the Markov inequality takes the form
P(X≥a)≤(a^c)E[X^5].

What is the value of c?

c= unanswered

Suppose that X_1,X_2,⋯ are random variables with mean E[X_i]=0 and the same finite variance var(X_i)=σ^2. Let M_n=(X_1+⋯+X_n)/n. In which of the following cases does M_n converge in probability to 0?

(i): X_1,X_2,⋯ are independent and identically distributed.

(ii): X_1,X_2,⋯ are identically distributed and cov(X_i,X_j)≤(1/2)^|i−j| for all i,j.

(iii): X_1,X_2,⋯ are such that cov(X_i,X_j)≤(1/2)^|i−j| for all i,j (but are not necessarily identically distributed).

(i)
(ii)
(iii)
(i)(ii)
(i)(ii)(iii)
unanswered

Let X be a Bernoulli random variable with unknown parameter p∈[0,1]. Does there exist an unbiased estimator Θ^ of 1/p? In other words, does there exist a function g such that Θ^=g(X) and that the E[Θ^]=1/p? (Recall that the function g cannot depend on p).
unanswered

Hint: E[Θ^]=E[g(X)]=(1−p)⋅g(0)+p⋅g(1).

Let X_i (for i≥1) be independent and identically distributed Poisson random variables with parameter λ=1. Let S_n=∑_{i=1}^n X_i. Compute the following limits:

lim_(n→∞)P(S_n≥n)= unanswered

lim_(n→∞)P(S_n≥n+sqrt(n))= unanswered

Answer 1

4.

lim_(n→∞)P(S_n≥n)= 0.5
lim_(n→∞)P(S_n≥n+sqrt(n)) = 1/(2*pi) = 0.159

Answer 2

1. c=-5

3. No

Answer 3

2. (i), (ii) and (iii)