STAT 330 Homework 3
Spring 2021
Due at 10:00pm EST on August 5th, 2021
1. [8] Let θ ∈ R be arbitrary. Show that the following statement is not always true:
If Xn
p→ θ, then lim
n→∞E[Xn] = θ.
Additionally, provide an intuitive explanation for why the above statement is not always true.
(Hint: let Zn ～ Bernoulli(p(n)), for some function p(·), and let Xn | Zn = 0 ～ N(θ, σ2/n),
and Xn | Zn = 1 ～ N(g(n), 1), for some function g(·). Choose g(·) and p(·) wisely.)
2. [8] Consider the following statement:
If Xn
d→ X and Yn d→ Y for non-constant random variables X and Y, then Xn+Yn d→ X+Y.
If this is a true statement, then prove it, and explain your intuition behind why the statement
is true. If it is not a true statement, then provide an example contradicting the statement,
and explain your intuition behind why the statement is not true.
3. [10] Suppose that X1, X2, . . . are independent normally distributed random variables. Let
θ1, θ2 ∈ R with θ1 6= θ2, and let 0 < p < 1. Define a sequence of real numbers μ1, μ2, . . . that
satisfy the following conditions:
(a) μi ∈ {θ1, θ2} for all i = 1, 2, . . . , (That is, the sequence of real nu
mbers μ1, μ2, . . . con-
tains only two numbers: θ1 and θ2.)
(b) lim
n→∞
|{i∈{1,2,…,n}:μi=θ1}|
n = p. (That is, for large enough n, the proportion of elements in
the set {μ1, μ2, . . . , μn} that are equal to θ1 is close to p.)
Suppose that E[Xi] = μi, and V ar[Xi] = σ
2 > 0 for all i = 1, 2, . . . I claim that s2 =
1
n 1
n∑
i=1
(Xi Xˉ)2 p→ τ2, for some quantity τ2. Find τ2. (A potentially helpful fact you can
use without proof: if Z ～ N(μ, σ2), then E[Z4] = μ4 + 6μ2σ2 + 3σ4.)
4. [8] Let θ > 0 be a constant. Let X1, X2, . . . be i.i.d. with pdf
f(x) =
{
θxθ 1, 0 < x < 1,
0, otherwise.
Find appropriate choices of an, b, and c such that an
((
n∏
i=1
Xi
) 1
n
b
)
d→ N(0, c).
5. [6] Let t1, . . . , tn be known constants. Suppose that X1, . . . , Xn are independent and Xi ～
Exponential(e βti) for all i = 1, 2, . . . , n.
(a) Write down the equation that you would need to solve to find the maximum likelihood
estimator of β.
(b) Let β be the MLE of β. Find appropriate choices of an, b, and c such that an(β b) d→
N(0, c).
(c) Let θi = E[Xi] for i = 1, 2, . . . , n. Find the MLEs of θi for all i = 1, 2, . . . , n.
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