Re: Existence of pdf and mgf
- From: Olw <anders_NOSPAM@xxxxxxxxxxx>
- Date: Sat, 28 Jan 2006 15:24:38 +0100
Robert Israel wrote:
In article <43da3f99$1@xxxxxxxxxxxxxxxxx>, Olw <anders_REMOVE@xxxxxxxxxxx> wrote:
I have some struggles understanding existence implications between the moment generating function and a probability distribution.
Going from a pdf / cdf to an MGF is OK, the MGF M(t)exists if the expectation E(e^{tx}) exist at some neighborhood of 0.
The other way, however. Given a function claiming to be an MGF, for example M(t) = t or M(t)=t/(1-t), for |t|<1. How can one know if it exists a corresponding pdf?
The MGF for imaginary t is the characteristic function phi(t) = E[exp(itX)]. According to Bochner's theorem, a function phi(t) is the characteristic function of a random
variable [in analyst's language, the Fourier transform of
a Borel probability measure on the real line] if and only if
it is continuous, positive definite, and phi(0) = 1.
Positive definite means for all real t_1, ..., t_n, the
n x n matrix with entries a_{j,k} = phi(t_j - t_k) is positive
semidefinite.
Also, say that one can decide that an MGF is valid, i.e., there is a corresponding pdf. Then the pdf can be found by inverse Laplace transforming M(-t). Is this in convenient also for distribution that can take negative values? Or does it exist any clever tricks for finding the pdf?
Fourier transform of the characteristic function.
Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
Robert,
Thank you!
Olw .
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