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

MAS328
(iii) What is the probability that 8 items are found anywhere within D(0, 3)∪
D(x, 3) with x = (7, 0) ?
This probability is
−9π (9π)8
e .
8!
Question 5. (20 marks)
Let (ξn)n≥0 be a two-state Markov chain on {0, 1} with transition matrix
0
1 − α
1
,
α
let (Nt)t∈R+ be a Poisson process with parameter λ > 0, and let the two-state
birth and death process Xt be defined by
Xt = ξNt, t ∈ R+.
(i) Compute the mean return time E[τ0 | X0 = 0] to 0 of Xt.
We have
and
hence
and
E[τ0 | X0 = 0] = 1
λ + E[τ0 | X0 = 1],
E[τ0 | X0 = 1] = 1
λ + αE[τ0 | X0 = 1],
E[τ0 | X0 = 1] =
E[τ0 | X0 = 0] =
5
1
λ(1 − α)
2 − α
λ(1 − α) .