# moment generating function

• Dec 8th 2008, 09:49 AM
gvidfhi
moment generating function
Hi i have a pdf in which i have to produce the mgf for.
now i understand what the mgf is and how to use it to produce E(X) and var(X) and so on. but i struggle with the actual process of creating it.

i would be very grateful if anyone could show a simple example for a continuous distribution explaining the steps involved

thanks
• Dec 8th 2008, 09:56 AM
Scopur
They key to doing the moment generating functions in my experience was to combine the $\displaystyle e^{tx}$ term with the rest of your function so it can be factor out of the integral or it can be turned into a recognizable integral. I am no master on them, as we only did them in class for some simple distributions, but if you told me which distribution you wanted help with I might be able to help.

Also check mathworld, there is tonnes of clarifications on how to derive MGF's.
• Dec 8th 2008, 12:22 PM
gvidfhi
it is a transformed exponential distribution

looks like this: f(x) = [x exp(-x^2)]/2

my main problem with mgf's is that i dont understand how to get rid of the x
• Dec 8th 2008, 03:34 PM
mr fantastic
Quote:

Originally Posted by gvidfhi
it is a transformed exponential distribution

looks like this: f(x) = [x exp(-x^2)]/2

my main problem with mgf's is that i dont understand how to get rid of the x

First of all, the pdf you've posted is wrong since it doesn't integrate to 1.

You probably meant f(x) = x exp(-[x^2]/2), or, using latex and being precise in the definition:

$\displaystyle f(x) = x e^{-x^2/2}$ for $\displaystyle x \geq 0$ and zero elsewhere.

Then $\displaystyle m(t) = E(e^{tX}) = \int_0^{+\infty} e^{tx} \, x \, e^{-x^2/2} \, dx$.

Note: The answer to this cannot be found in terms of elementary functions. Where has your transformed pdf come from? Why do you require its mgf?