find the pdf of a moment generating function?

Question

Let Mx(t)=e^t/6+e^2t/3+e^3t/2 be the moment generating function of X. Find the pdf of X.

Where do I go from here?

Answer 1

Given any moment generating function M(t), M(-t) is the two sided or bilateral Laplace transform of the density function[1]. For the bilateral Laplace transform, the integration extends from - infinity to + infinity whereas the unilateral (one sided) Laplace transform is 0 to + infinity [2].

Therefore assuming M(-t) is a Laplace transform will result in the density function having a step function at 0. There are transformations between bilateral and unilateral Laplace transforms so unilateral inverse transforms can be used[3]. You will need to work through some derivations to get the correct transform relationships.

The inverse Laplace transform of M(-t) will generate the density function. Be sure to perform the actual Inverse Laplace transform and not the integral given above. Because of the properties of Laplace transforms, you should be able to perform the transforms on each term in M[t]. If you are using Mathematica, these transforms can be done directly.

Then integrating the resulting density function from - Infinity to x will give the cumulative distirbution function F(x).

You can check that the result gives the original moment generation function by integrating:

M(t) = Int_{- infinity to + infinity}(e^(x t) F'(x) d x)

F'(x) is the PDF. This should get you within a constant factor of the original function. If off look at the bilateral to unilateral relationships.

If you are using Mathematica, I can send you a work sheet if you email me a request. If this is a homework assignment, you will have to work through this anyways to understand your result.

Answer 2

C'mon buddy, that's easy. As you know the moment generating function is the Laplace transform of the density function. So,the density function is the inverse Laplace transform of the moment generating function. And then the PDF is just the integrate of the density function .

So you must calculate:

f(w) = int_ {from 0 to infinity} (e^t/6+e^2t/3+e^3t/2) e^{-tw} dt

and then

F(x) = int_{from -infinity to x} f(w) dw

where f is the density function and F is the PDF.

Answer 3

http://www.wolfram.mathworld.com