Mathematical Probability (STAT2003/STAT7003)
Problem Set 3
The due date/time is given on Blackboard. STAT7003 students have an additional ques-
tions [2(f), 3(e) and 4(c)] marked with a star (*).
1. Let X1 and X2 be two independent random variables with the probability density
function of Xi given by
f(x) =
⇢
1
x ln 2 , x 2 (0.5, 1)
0, else.
(a) Determine the probability density function of Y = X1X2. [6 marks]
(b) Determine the probability density function of
Z =
⇢
Y, Y 2 (0.5, 1)
2Y, Y 2 (0.25, 0.5]
[3 marks]
2. (a) Let U and V be two independent random variables such that U ⇠ Ber(1 %)
with % 2 [0, 1) and V ⇠ Exp() with > 0. Determine the moment generating
function of W = UV . [3 marks]
(b) Let {Wi}1i=1 be a sequence of independent and identically distributed random
variables where Wi has the same distribution as W from part (a). Let X0 be a
random variable independent of the {Wi}1i=1 and define the sequence of random
variables {Xn}1n=0 recursively as
Xn = %Xn1 +Wn.
Show that if X0 ⇠ Exp(), then Xn ⇠ Exp() for all n > 0. [3 marks]
(c) Determine Cov(Xn+k, Xn) for all n and k > 0. [2 marks]
(d) Determine E(Xn |Xn1 = x) and Var(Xn |Xn1 = x). [2 marks]
(e) Simulate a sample path for n = 0, . . . , 50 with % = 0.75 and = 1. [2 marks]
(f) * Let {Yn}1n=0 be the sequence of random variables such that Y0 ⇠ Gamma(2,)
for some > 0, and
Yn = %Yn1 +fWn,
where {fWn}1n=1 is a sequence of independent and identically distributed ran-
dom variables and Y0 is independent of the {fWn}1n=1. Determine the moment
generating function of the {fWn}1n=1 such that Yn ⇠ Gamma(2,) for all n > 0.
Describe how fWn can be simulated using random variables with Bernoulli and
Exponential distributions. [3 marks]
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3. Consider the following system comprised of three components:
The system is working if there is a path from left to right through working compo-
nents. Components fail independently and the time to failure for each component
has an exponential distribution with a mean of one year.
(a) Determine an expression for the probability that the system is working at time
t. [3 marks]
(b) Determine the mean time to failure for the system. [2 marks]
(c) Determine the probability that component two in the system is still working
at time t given the system is working at time t. What is the limiting value as
t!1? [3 marks]
(d) Determine the failure rate for the system. [1 mark]
(e) * Show the system has an increasing failure rate. [2 marks]
4. Consider the 4-state Markov chain X = {X1, X2, . . .} described by the following
transition graph.
1 2 3 412
1
2 1 ↵
↵
1

1

(a) Determine (↵, , ) such that the limiting distribution of theX is ⇡ = ( 110 ,
2
10 ,
3
10 ,
4
10).
[4 marks]
2
(b) Let (↵, , ) = (13 ,
1
4 ,
1
5). Using a grid like the one below, sketch by hand a
typical realisation of Xn, n = 1, . . . , 30, where X1 = 1. [2 marks]
5 10 15 20 25 30
1
2
3
4
(c) * For (↵, , ) = (12 ,
1
2 ,
1
2) the limiting distribution of the Markov chain X is
= (14 ,
1
4 ,
1
4 ,
1
4). Define the sequence of random variables {Yn}10n=1 such that Yn =
X11n. Show that Y = {Y1, Y2, . . . , Y10} is a Markov chain and determine the
matrix of one-step transition probabilities for Y . [5 marks]
5. Let X1, . . . , Xn be independent random variables where the Xi have a Geo(p) distri-
bution. Define Sn = X1 +X2 + · · ·+Xn.
(a) Show that for any a > 1 and any t 2 [0, ln(1 p)),
P(Sn > an/p) 6
pnen(1a/p)t
(1 (1 p)et)n =: H(t; a).
[2 marks]
(b) As this upper bound H(t; a) holds for all t 2 [0, ln(1 p)), the tightest upper
bound is found by minimising H(t; a) over t. For a fixed value of a, find the
value ta which minimises H(t; a). [2 marks]
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