Suppose that {N(t), t ≥ 0} is a Poisson process with rate λ > 0 and that S is a random variable having a uniform distribution on the interval [0,2]. (a) Obtain the moment-generating function of the...

Suppose that
{N(t), t
≥ 0} is a Poisson process with rate λ > 0 and that S is a random variable having a uniform distribution on the interval [0,2].

(a) Obtain the moment-generating function of the random variable
N{t
+
S). Indication.
If
X
has a Poisson distribution with parameter a, then
Mx(t)
=
exp{α(e^t
— 1)}.

(b) Calculate the mean and the variance of
N{t + S).