(18pts)

Suppose that

T

is a continuous random variable with support

(0,\\\\infty )

.\ expected value

\\\\mu

, variance

\\\\sigma ^(2)

and

MGFM(t).,T

is independent of the\ Poisson process

{N(t),t>0}

with rate parameter

\\\\lambda

. Consider the random\ variable

N(T)

. Find the following in terms of

\\\\mu ,\\\\sigma ^(2)

and

M(t)

:\ (a)

E[N(T)]=\\\\lambda \\\\mu

\ (b)

Var[N(T)]=\\\\lambda \\\\mu +\\\\lambda ^(2)\\\\sigma ^(2)

\ (c)

Cov[N(T),T]=\\\\lambda \\\\sigma ^(2)

\ (d)

P(N(T)>0)=1-M_(T)(-\\\\lambda )

4. (18pts) Suppose that T is a continuous random variable with support (0,). expected value , variance 2 and MGFM(t). T is independent of the Poisson process {N(t),t>0} with rate parameter . Consider the random variable N(T). Find the following in terms of .2 and M(t) : (a) E[N(T)]= (b) Var[N(T)]=+22 (c) Cov[N(T),T]=2 (d) P(N(T)>0)=1MT()