MAS328 Solutions to the final exam. Question 1. (20 marks) Four ...

(ii) Compute the probability generating function G(s) of X.
The probability generating function of X is given by
G(s) = E[s X ] = 1
2
∞
k=0
(s/2) k = 1
2 − s .
MAS328
(iii) What is the probability that the husband’s male line of descent will
cease to exist by the third generation ?
The probability we are looking for is
G(G(G(0))) =
1
2 − 1
2−1/2
= 3
4 .
Question 4. (20 marks)
Suppose that X(A) is a spatial Poisson process of discrete items scattered
on the plane R 2 with intensity λ = 0.5 per square meter. No evaluation of
numerical expressions is required in this question.
(i) What is the probability that 10 items are found within the disk D(0, 3)
with radius 3 meters centered at the origin ?
This probability is
−9π/2 (9π/2)10
e .
10!
(ii) What is the probability that 5 items are found within the disk D(0, 3)
and 3 items are found within the disk D(x, 3) with x = (7, 0) ?
This probability is
−9π/2 (9π/2)5
e
5!
4
(9π/2)3
× e−9π/2 .
3!