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PRIFYSGOL ABERYSTWYTH UNIVERSITY
INSTITUTE OF MATHEMATICS AND PHYSICS
SEMESTER 2 EXAMINATIONS, JANUARY 2009
MA37410 - Probability and Stochastic Processes
Time allowed - 2 hours
All questions may be attempted.
Marks gained from questions in section B will be given greater consid- eration in assessing a Örst class performance.
Calculators are permitted, provided they are silent, self-powered, with- out communications facilities, and incapable of holding text or other material that could be used to give a candidate an unfair advantage. They must be made available on request for inspection by invigilators, who are authorised to remove any suspect calculators.
Statistical tables will be provided.
You may assume without proof that (a) if X Poisson, P 0 (), then X has probability generating function
GX (s) = expf(s 1)g;
(b) if Y Normal, N(, ^2 ); then Y has moment generating function
MY () = expf + 12 ^2 ^2 g;
(c) if cn! c; as n! 1, then (1 + c nn )n^ ! ec:
Section A
- If X and Y have joint probability density function (pdf)
f (x; y) =
x^3 y^3
; x > 1 ; 1 < y < x;
(and 0 elsewhere), sketch the region where f (x; y) is positive.
(a) Show that the marginal densities of X and of Y are respectively
f (x) =
4(x^2 1) x^5
; x > 1 and f (y) =
y^5
; y > 1 :
[7]
(b) Calculate EX and EY. [5]
(c) Show that E[XjY = y] = 2y and verify that EX = EE[XjY ]. [8]
- The number of accidents on the M470 motorway in a month is a random variable N with mean 4 and standard deviation 2. The ithaccident involves a random number of vehicles Vi which has mean 2 and variance
- The Vi are independent of each other and of N. If the total number of vehicles involved in all N accidents is T = T (N ) = V 1 + V 2 + ::: + VN ; Önd (i) E[T jN ]; (ii) Var[T jN ] and hence (iii) Var[T ]: [10]
- The times T 1 ; T 2 ; ::::; Tn taken by a call centre worker to handle enquiries are independent random variables with a standard deviation 5 where is the mean time.
(a) What is the approximate distribution of S, the total of a large number n of these times? Name the theorem which justiÖes your answer. (b) How large must n be so that we can be 90% certain that the relative error in estimating , i.e. T^ ^ ; is less than 1.6? [10]
Section B
- (a) Let T =
PN
i=1 Ri^ where^ N^ is an independent, integer-valued ran- dom variable having probability generating function (pgf) GN (s) and R 1 ; R 2 ; R 3 ::::are independent and identically distributed dis- crete random variables whose pgf is GR(s). If N is independent of the Ri, show that the pgf of T is
GT (s) = GN (GR(s)):
[5]
(b) A machine tool representative makes sales to a random number of companies, N , each week, where N has the Poisson distribution P 0 () with = ln 3 : At company i he sells Ri machine tools where Ri has probability generating function (pgf)
GR(s) = ln(1
2 s 3
)= ln 3; jsj <
Show that if N; R 1 ; R 2 ; ::::::are all independent, then the pgf of T , the total number of sales in a week, is
GT (s) = [3 2 s] ^1 :
[4]
(c) By expanding the pgf of T + 1, Önd its probability mass function and identify its distribution by name. [8]
- If X 1 ; X 2 ; :::; Xn; :::are independent with the same moment generating function (mgf) M () = [1 3 ^2 ] ^1 ;
Önd the mgf of Un =
Pn i=1 Xi=
p n and hence its asymptotic distribu- tion. [8]
- A branching process starts with a single individual in the zeroth^ gen- eration and has a family size distribution with probability generating function (pgf) G(s). Let Xn be the size of the nth^ generation.
(a) State the relationship between Gn(s), the pgf of Xn, and Gn+1(s): If G(s) = (1 + 3s)=(5 s); show that
Gn(s) =
n (n 4)s 4 + n(1 s)
[8]
(b) Show that the probability that the nth^ generation is the Örst one at which the population becomes extinct is 4 (n + 4)(n + 3)
[6]
- A Markov Chain has transition matrix
2 6 6 6 6 6 6 4 0 : 4 0 : 2 0 0 : 4 0 0 0 0 : 1 0 0 0 : 9 0 0 0 0 : 2 0 0 0 : 8 0 0 0 : 6 0 : 4 0 0 0 0 : 5 0 0 0 : 5 0 0 0 0 : 4 0 : 3 0 0 : 3
(a) Find the irreducible classes, say which classes are recurrent or transient and state the period of each class. Reorder the states so as to display the class structure of the chain. [10] [In the remainder of the question you may, if you wish, use the re-ordered states.] (b) For any recurrent classes you Önd, calculate the stationary distri- bution. [8] (c) For each transient state i and every recurrent class C calculate iC , the long-run probability of entering C, given that the process started in i. [3]
