# Moment generating function and distribution function?

• Jan 25th 2009, 04:54 AM
DCU
Moment generating function and distribution function?
http://img212.imageshack.us/img212/4338/yeahou8.jpg

I got part a out easily enough. I'm having big problems on how to do part b and c. I'm like this (Headbang). Thanks for any help
• Jan 25th 2009, 01:05 PM
mr fantastic
Quote:

Originally Posted by DCU
http://img212.imageshack.us/img212/4338/yeahou8.jpg

I got part a out easily enough. I'm having big problems on how to do part b and c. I'm like this (Headbang). Thanks for any help

(b) I suppose they want you to calculate the cdf of X: $\displaystyle F(x) = \int_0^x \lambda e^{-\lambda u} \, du$.

This should be a simple integral for you to calculate.

(c) Apply the definition: $\displaystyle M_X (t) = E\left(e^{tX}\right) = \int_0^{+ \infty} e^{tx} \lambda e^{-\lambda x} \, dx = \lambda \int_0^{+ \infty}e^{-(\lambda - t) x} \, dx$.

Again, this should be a simple integral for you to calculate. What happens if $\displaystyle \lambda - t < 0$ ....?
• Jan 25th 2009, 01:42 PM
DCU
Quote:

Originally Posted by mr fantastic
(b) I suppose they want you to calculate the cdf of X: $\displaystyle F(x) = \int_0^x \lambda e^{-\lambda u} \, du$.

This should be a simple integral for you to calculate.

(c) Apply the definition: $\displaystyle M_X (t) = E\left(e^{tX}\right) = \int_0^{+ \infty} e^{tx} \lambda e^{-\lambda x} \, dx = \lambda \int_0^{+ \infty}e^{-(\lambda - t) x} \, dx$.

Again, this should be a simple integral for you to calculate. What happens if $\displaystyle \lambda - t < 0$ ....?

if $\displaystyle \lambda - t < 0$, the value of the integral will go to infinity?
• Jan 25th 2009, 01:57 PM
mr fantastic
Quote:

Originally Posted by DCU
if $\displaystyle \lambda - t < 0$, the value of the integral will go to infinity?

So that tells you the answer to the last part of (c).
• Jan 25th 2009, 02:02 PM
DCU
Quote:

Originally Posted by mr fantastic
So that tells you the answer to the last part of (c).

so when lambda is greater than t?