# Complex integration of a moment generating function

• May 29th 2008, 10:08 AM
langweilig
Complex integration of a moment generating function
Hi,
I have a moment generating function (infinite series). Supposedly, I can recover the probability distribution by doing the following integration:
f(x) = int m(t)exp(-tx) dt /(2ipi),
the integration going from c-i*infinity to c+i*infinity.

As an example, the moment generating function for a gaussian is exp(mu*t+sigma^2*t^2*1/2). Can someone show me how to do this integration? Especially if the mgf is expanded in a taylor series...THANKS!
• May 29th 2008, 04:36 PM
mr fantastic
Quote:

Originally Posted by langweilig
Hi,
I have a moment generating function (infinite series). Supposedly, I can recover the probability distribution by doing the following integration:
f(x) = int m(t)exp(-tx) dt /(2ipi),
the integration going from c-i*infinity to c+i*infinity.

As an example, the moment generating function for a gaussian is exp(mu*t+sigma^2*t^2*1/2). Can someone show me how to do this integration? Especially if the mgf is expanded in a taylor series...THANKS!

There's a connection between the moment generating function and the Laplace transform. I have no time to elaborate now. You might have a read of this: http://www.bibalex.org/Supercourse/S...0001/19461.pdf