?A random variable X has the gamma distribution \(f(x)=\frac{\lambda}{\Gamma(r)}(\lambda

Chapter 5, Problem 5.8.8

(choose chapter or problem)

A random variable X has the gamma distribution

              \(f(x)=\frac{\lambda}{\Gamma(r)}(\lambda x)^{r-1} e^{-\lambda x}\), x>0

(a) Show that the moment-generating function for \(t<\lambda\) of X is

                                        \(M_{X}(t)=\left(1-\frac{t}{\lambda}\right)^{-r}\)

(b) Find the mean and variance of X.

Text Transcription:

f(x)=lambda/Gamma(r)(lambda x)^r-1 e^-lambda x

t<lambda

M_X(t)=(1-t/lambda)^-r