Problem 5. Arrivals during overlapping time intervals

Question

Consider a Poisson process with rate lambda.
Let N be the number of arrivals in the interval from 0 to t.
Let M be the number of arrivals in the interval from 0 to (t+s).
t is greater than 0,
s is greater or equal to 0

1. For numbers n and m that satisfy the constraint that 0 is less than n which is less than m, the conditional PMF p_M|N(m|n) of M given N is of the form a/(b factorial) for suitable algebraic expressions in place of a and b.

a = ?
b = ?

2. E[N*M] = ?

Answer 1

a : (lambda*s)^(m-n)*e^(-lambda*s)
b : m-n
2 : (lambda*t)*(lambda*s)+lambda*t+(lambda*t)^2