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

Discussion in 'Probability' started by Rueful Rabbit, Nov 20, 2011.

1. ### Rueful RabbitGuest

Hi All,

I have an MGF and I want to find its PDF.
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.

Rueful Rabbit, Nov 20, 2011

2. ### Herman RubinGuest

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.

Herman Rubin, Nov 21, 2011