Q's 1)How to find PDF from MGF?; 2)A problem

Hi All,

I have an MGF and I want to find its PDF.
The advice at http://mathforum.org/library/drmath/view/74480.html
is ...
f(x) = 1/(2pi) * integral from -infinity to infinity of M_Y(it) * exp(-
itx) dt

But I could not find any other reference. Know or any good links?

But I hear you ask, "Why dear fool would you want such a thing?"

Well, I'm given the following MGF ...
M_w(t) = (1-2t)^-5
What is P[W > 11.78]?

I calculate the E[X] (mean) = M_w'(0) = 10
However, I need to know more than this about the distribution than
simply calculating moments, don't I?

I looked at this table in Wikipedia ... http://en.wikipedia.org/wiki/Moment-generating_function
and the given MGF looks (to me) like a chi-squared(10) (I'm not hot on
chi-squares yet)

1) Do U need to convert to a pdf for this problem?
2) Are there standard tables?
3) If 1) or 2) not relevant to my problem, nudge me in the right
direction.

Thanks, James.

 

It is a chi-squared with 10 (2x5) degrees of freedom.
It is not difficult to integrate the density.

In this case, a few moments work well, as it is approximately normal.
The formula you quote is for the density; there is another for
the cdf. However, that is for numerical work.

In general, inverting a moment generating function, unless
it is that of a known distribution, is only possible numerically,
and is not often done.