MATH 452/STAT 552
Midterm Supplementary Questions
October 9, 2020
Your preparation should ALSO include the other materials of this course.
Answer keys can be found in the next page.
Question 1. Suppose that the lifetimes of the light bulbs purchased from a hardware
store are i.i.d. distributed as e, where e is an exponential random variable with a
mean of 3 months. Each month a new light bulb is purchased and replaces the
previous one, but the budget can only support one light bulb in each month. What
is the probability that within 6 months, all the light bulbs burn out before they are
replaced?
Question 2. Let (Y1, Y2) be joint normal random variables with E[Y1] = E[Y2] = 0,
E[Y 21 ] = 2, E[Y 22 ] = 5, and Cov(Y1, Y2) = 3. Show that the following joint normal
random variables are independent: X1 = 2Y1 − Y2 and X2 = −Y1 + Y2.
Question 3. Let X be an exponential random variable with mean 1/2. Let Y = 1
if X < 2 and Y = 2 if X ≥ 2. Find P(0 ≤ X ≤ 1|Y = 1).
Question 4. Let (Xn) be a Markov chain with state space {1, 2, 3}. Suppose that
P(X3 = 1|X0 = 1) = 98216 , P(X3 = 2|X0 = 1) = 44216 , P(X3 = 3|X0 = 1) = 74216 ,
P(X2 = 2|X0 = 1) = 436 , P(X2 = 2|X0 = 2) = 1236 , P(X2 = 2|X0 = 3) = 636 .
Find P(X5 = 2|X0 = 1).
Question 5. Consider the following transition probability matrix:
P =

a b c
a 1/3 1/3 1/3
b 1/2 1/2 0
c 1 0 0
.
(1) Find Pa(X1 = a,X2 = b,X3 = a).
(2) Is this transition matrix irreducible?
(3) Show that pi = (1/2, 1/3, 1/6) is a stationary distribution of P .
1
Answers
Question 1. (1− e− 13 )6.
Question 2. Since X1, X2 are joint normal and Cov(X1, X2) = 0, the required inde-
pendence follows.
Question 3. (1− e−2)/(1− e−4).
Question 4. 1364/7776 = 341/1944 ≈ 0.175411.
Question 5. (1) 1/3 · 1/3 · 1/2. (2) Yes. (3) Show that pi satisfies piP = pi.
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