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STAT2003/STAT7003 Mathematical Probability Problem Set 3
发布时间：2022-05-25
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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) = 
(a) Determine the probability density function of Y = X1X2 . [6 marks]
(b) Determine the probability density function of
Z = 
[3 marks]
2. (a) Let U and V be two independent random variables such that U ∼ Ber(1 − ϱ) with ϱ ∈ [0, 1) and V ∼ Exp(λ) with λ > 0. Determine the moment generating function of W = UV . [3 marks]
(b) Let {Wi }
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 }
and define the sequence of random variables {Xn }
recursively as
Xn = ϱXn −1 + Wn .
Show that if X0 ∼ Exp(λ), then Xn ∼ Exp(λ) for all n ⩾ 0.
(c) Determine Cov(Xn+k,Xn ) for all n and k ⩾ 0.
(d) Determine E(Xn |Xn −1 = x) and Var(Xn |Xn −1 = x). (e) Simulate a sample path for n = 0, . . . , 50 with ϱ = 0.75 and λ = 1.
[3 marks] [2 marks] [2 marks] [2 marks]
(f) * Let {Yn }
be the sequence of random variables such that Y0 ∼ Gamma(2,λ)
for some λ > 0, and
Yn = ϱYn −1 + Wn ,
dom variables and Y0 is independent of the {Wn }
. Determine the moment
generating function of the {Wn }
such that Yn ∼ Gamma(2,λ) for all n ⩾ 0.
[3 marks]
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
.
(b) Determine the mean time to failure for the system.
[3 marks]
[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 → ∞? [3 marks]
(d) Determine the failure rate for the system.
(e) * Show the system has an increasing failure rate.
[1 mark]
[2 marks]
4. Consider the 4-state Markov chain X = {X1 ,X2 , . . .} described by the following transition graph.
γ
1
2
1 − γ
(a) Determine (α,β,γ) such that the limiting distribution of the X is π = (
, 
, 
, 
). [4 marks]
(b) Let (α,β,γ) = (
, 
, 
). 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
(c) * For (α,β,γ) = (
, 
, 
) the limiting distribution of the Markov chain X is = ( 
, 
, 
, 
). Define the sequence of random variables {Yn }
such that Yn = X11−n . 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 ∈ [0, −ln(1 − p)),
P(Sn ⩾ an/p) ⩽ 
=: H(t;a).
[2 marks]
(b) As this upper bound H(t;a) holds for all t ∈ [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]
