X and Y are independent exponential random variables with me

X and Y are independent exponential random variables with mean µ_X and µ_Y respectively.

Find the MGF of Z = X Y

Solution

Let \'f\' be the MGF of X.

Let \'g\' be the MGF of Y.

then f(X)=E[e^X]=µX

g(Y)=E[e^y]=µY

then, MGF(Z)=MGF(X-Y)= f(X)*g(-Y)=(µX)*(-µY)

MGF(Z)=(-µ²XY)