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 numbers 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 − X¯)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 βˆ be the MLE of β. Find appropriate choices of an, b, and c such that an(βˆ − 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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