Moment generating function and distribution function?

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January 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
January 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: $F(x) = \int_0^x \lambda e^{-\lambda u} \, du$ .

This should be a simple integral for you to calculate.

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

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

Quote:

Originally Posted by mr fantastic

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

This should be a simple integral for you to calculate.

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

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

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

Quote:

Originally Posted by DCU

if $\lambda - t < 0$ , the value of the integral will go to infinity?

So that tells you the answer to the last part of (c).
January 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?