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

Chapter 5, Problem 5.8.7

(choose chapter or problem)

A random variable X has the exponential distribution

                                       \(f(x)=\lambda e^{-\lambda x}\), x>0

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

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

(b) Find the mean and variance of X.

Text Transcription:

f(x)=lambda e^-lambda x

t<lambda

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