Let X UV U is exponentially distributed with mean 2 and V

Let X = U+V, U is exponentially distributed with mean = 2 and V is a Poisson random variable with mean 3, U and V are independent. Find Mx(t).

Solution

Let

E(U)= 1/r

which is the known mean for an exponential function,

then In the context of the Poisson process, the parameter r is known as the rate of the process. On average, there are 1 r time units between arrivals, so the arrivals come at an average rate of r per unit time.

Now

Mx(t)= 1/r + r_p/2

= 2+ 3/2

2.5