MATH 452/STAT 552
Final Exam Supplementary Questions
December 4, 2020
The questions below are only based on the materials after the midterm,
but the final exam is comprehensive.
Question 1 (Continuous-time Markov chains). Consider the Q-matrix
Q =

a b c
a −4 2 2
b 3 −6 3
c 4 4 −8
.
(1) Find the parameters λi of the holding times of all states and the transition
probability matrix (Pij) of the embedded Markov chain (Pii = 0 for all i).
(2) Find the stationary distribution of the embedded Markov chain from (1).
(3) Is the probability vector pi = ( 6
13
, 4
13
, 3
13
) a stationary distribution of the continuous-
time Markov chain with Q-matrix given above? Justify your answer.
Question 2 (Poisson processes). Let e1, e2 be independent exponential random
variables such that E[ei] = λ−1i . Find E[e2|e1 < e2]
Question 3 (Renewal processes). Suppose that the interarrival distribution of a
renewal process (Nt; t ≥ 0) is the continuous uniform distribution on [0, 2].
(1) Show that the renewal functionm(t) = E[N(t)] satisfiesm(t) = t∧2
2
+1
2
∫ t∧2
0
m(s)ds,
where a ∧ b = min{a, b}.
(2) Find limt→∞Nt/t.
Question 4 (Brownian motion and Gaussian stochastic integrals). Let B =
(Bt; t ≥ 0) be a standard Brownian motion.
(1) Find E[B22B21 ]. (Recall that if Z ∼ N (0, σ2), then E[Z4] = 3σ4.)
(2) Let It = I0 +
∫ t
0
rdBr, where I0 ∼ N (0, 2) is independent of B. Find E[It] and
Var(It).
Question 5 (The strong law of large numbers and the central limit theo-
rem). Let X1, X2, · · · , Xn, · · · be i.i.d. exponential random variables with mean 2.
Write out the following limits explicitly for the sums of independent random variables
Sn =
∑n
j=1Xj:
(1) The strong law of large numbers.
(2) The central limit theorem.
1
Answers
Question 1. (1) λa = 4, λb = 6, λc = 8.
P =

a b c
a 0 1/2 1/2
b 1/2 0 1/2
c 1/2 1/2 0
.
(2) The embedded Markov chain is the random walk on a graph. The graph is the
discrete cycle of 3 vertices where each vertex has 2 neighbors. Hence, the stationary
distribution is pi(a) = pi(b) = pi(c) = 2/6. (3) Yes. piQ = 0.
Question 2. 1
λ1+λ2
+ 1
λ2
.
Question 3. (1) The required equation is the renewal equation with f(t) = 1
2
1[0,2](t).
(2) 1.
Question 4. (1) 4. (2) E[It] = 0, Var(It) = 2 + 13t
3.
Question 5. (1) P(limn→∞ Sn/n = 2) = 1.
(2) limn→∞ P(Sn−2n√4n ≤ x) = 1√2pi
∫ x
−∞ e
−t2/2dt for all x ∈ R.
2

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