?A random variable X has the Poisson distribution \(f(x)=\frac{e^{-\lambda}

Chapter 5, Problem 5.8.2

(choose chapter or problem)

A random variable X has the Poisson distribution

                      \(f(x)=\frac{e^{-\lambda} \lambda^{x}}{x !}\), \(x=0,1, \ldots\)

(a) Show that the moment-generating function is

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

(b) Use \(M_{X}(t)\) to find the mean and variance of the Poisson random variable.

Text Transcription:

f(x)=e^-lambda lambda^x/x!

(x=0,1, ldots

M_X(t)=e^lambda(e^t-1)

M_X(t)