Stat-461/Mast-729G/4 (2020-21)
Sample questions for final exam
1. Suppose that a store opens at 0pm and customers arrive at the store according
to a non-homogeneous Poisson process {N(t), 0 ≤ t ≤ 8} with the intensity
function
λ(t) =
{
4− t
4
, 0 ≤ t ≤ 4,
3 + t−4
4
, 4 < t ≤ 8.
Let Sj denote the time at which the j-th customer arrives, j ≥ 1. Compute
(a) P [S11 ≤ 5|N(3.5) = 9].
(b) P [S2 ≤ 4|N(6) = 4].
2. Suppose N(t) is a Poisson process with rate function λ(t) = | sin(t)|, t ≥ 0.
Denote the event-times of N(·) by 0 < S1 < S2 < · · · . Find P [S1 ≤ pi < S3],
and generate the first two event-times S1, S2.
3. Let
N = min
{
n ≥ 1 :
∣∣∣∣U21 + · · ·+ U2nn − 0.333
∣∣∣∣ < 0.001} ,
where U1, U2, . . . are iid Uniform (0, 1) random variables. Estimate E(1/N)
by k−1
∑k
i=1(1/Ni) where k ≥ 200 and sd(1/N)/sqrt(k)< 0.004.
4. For independent sequences of iid Uniform (0, 1) random variables {U1, U2, . . .}, {V1, V2, . . .}
define
N = min
{
n : Un ≤ 1
Vn + 1
}
.
Obtain E(N) and joint as well as marginal pdf’s of (UN , VN).
5. (a) Suppose
P (X = i) = 0.2 for i = 1, . . . , 5.
We want to generate random numbers from X by the acceptance-rejection
method using
P (Y = i) = p(1− p)i−1 for i = 1, 2, . . . .
for some 0 < p < 1. Find p that minimizes(
max
i
P (X = i)
P (Y = i)
)
and generate a value of X using that p.
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(b) Suppose
P (X = i) =
i
15
for i = 1, . . . , 5.
We want to generate random numbers from X by the acceptance-rejection
method using
P (Y = i) = p(1− p)i−1 for i = 1, 2, . . . .
for some 0 < p < 1. Find p that minimizes(
max
i
P (X = i)
P (Y = i)
)
and generate a value of X using that p.
6. Let X ∼ Geometric (1/2) and Y ∼ Geometric (1/3) be two independent
random variables. Obtain the conditional distribution of X, given that X =
Y +1, and then generate a value of a random variable having this conditional
distribution.
7. Generate a value of a random variable X with p.d.f.
f(x) = xe−x
2
+
1
2
e−x, x ≥ 0,
using the indicated methods:
(a) Use the acceptance/rejection method, with g(x) = e−x, x ≥ 0, distribu-
tion as g.
(b) Use the composition method.
8. Generate a value of a random vector (X1, X2) with joint pdf
f(x1, x2) = 3(x1 + x2), 0 < x1, 0 < x2, x1 + x2 ≤ 1,
by acceptance-rejection method, using another random vector (Y1, Y2) with
joint pdf
g(y1, y2) =
1
4
√
y1y2
, 0 < yi ≤ 1, i = 1, 2.
What is the expected number of rejections?
9. Suppose (X1, X2) have the joint pdf f(x1, x2) = 2(x1+x2)I(0 ≤ x1 < x2 ≤ 1).
Find the marginal pdf f2(x2) of X2, the conditional pdf f(x1|x2) of X1, given
X2 = x2, and hence, or otherwise, generate a value of (X1, X2).
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10. Consider a single-server queueing model where customers arrive according to
a homogeneous Poisson process with rate λA = 10. Upon arrival a customer
either enters service if the server is free at that moment or else joins the waiting
queue if the server is busy. When the server completes serving a customer it
then begins serving the customer that had been waiting the longest. If there are
no waiting customers the server remains idle until the next customer arrives.
The service times are i.i.d. Exponential random variables with rate λS = 12,
independent of arrivals.
(a) Generate the arrival times of the first three customers and their departure
times.
(b) Generate the arrival times of the first three customers and their departure
times, given that both Customer-2 and Customer-3 have tolerance times
which are i.i.d. Exponential (15) random variables, i.e., each will leave
after an Exponential (15) amount of time from his/her arrival, if not
served by then.
11. Consider a queue with 2 parallel servers, where server i takes Exponential (i)
amount of time to complete a service, i = 1, 2. Services are independent of one
another and of the arrivals which occur according to a homogeneous Poisson
process with rate 2. Every arrival goes to the idle server if one of the servers
is busy, to server 1 if both servers are free and wait in a common queue if both
servers are busy; the one that has been waiting the longest is served first when
a server becomes idle. Simulate the arrival and departure times of the first 3
arrivals, and calculate the idle times of the two servers until the departure of
the 3rd arrival.
12. Suppose that X has the pmf:
P (X = x) = 3
∫ 1
0
y(1− y)x+1dy, x = 1, 2 . . . .
Generate a value of X by the composition method and also a value of an
antithetic variable X ′ for X.
13. Suppose that (X,Y ) have the joint p.d.f.
f(x, y) =
{
2
√
x, if 0 ≤ x ≤ 1 and 0 < y < √x,
0, otherwise.
We want to use an estimator of the form [Z + c(X − E(X))] to estimate
P{X ≤ 2Y }, where
Z =
{
1, if X ≤ 2Y,
0, otherwise.
3
Find the (theoretical) optimal choice of c.
14. Suppose that (X,Y ) have the joint p.d.f.
f(x, y) = 4e−
x
y
−2y2 , x > 0, y > 0.
We want to estimate θ = P{|X − Y | < 1}. Consider
Z =
{
1, if |X − Y | < 1,
0, otherwise.
(a) Obtain g(Y ) = E(Z|Y ).
(b) Compute
∑2
i=1 g(Yi)/2 by generating Y1 and Y2.
15. Let X = eU , where U is a Uniform (0, 1) random variable.
(a) Compute Var((X +X ′)/2), where X ′ is antithetic to X.
(b) Obtain c that minimizes Var(X + c(U − 0.5)).
16. Suppose that Y is an Exponential (1/2) random variable, and conditional on
Y = y, N is a Poisson random variable with mean y. We want to estimate
p = P{2 ≤ N ≤ 3}. Obtain E(Z|Y ), where
Z =
{
1, if 2 ≤ N ≤ 3,
0, otherwise.
Compute the estimate
∑5
i=1E(Zi|Yi)/5 by generating 5 i.i.d. copies Y1, . . . , Y5
of Y .
Values of Uniform (0, 1) Random Variables
0.81222166 0.05464751 0.06930601 0.94253943 0.79179519
0.49573802 0.96771072 0.89496062 0.89866417 0.85499261
0.67242223 0.33383212 0.08998713 0.67070207 0.32724888
0.24365016 0.14752814 0.57458281 0.82379935 0.48721456
0.37257331 0.22673586 0.22982463 0.96236055 0.87142464
0.64248448 0.15947970 0.31127181 0.37530767 0.46249647
0.22759288 0.61511912 0.42639104 0.00558105 0.57204456
0.78462924 0.83509543 0.67738296 0.99870467 0.29117284
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