MaplePrimes - answers and comments on Question, How do I find the moment generating function from a given PDF?

It can be done with Maple

Markiyan Hirnyk — Thu, 08 Nov 2012 22:22:22 Z

Up to Wiki (see http://en.wikipedia.org/wiki/Moment-generating_function ),
the moment-generating function M[X](t) of an absolutely continuous random variable X
can be calculated in such a way
M[X](t):=unapply(int(exp(t*u)*PDF(X,u),u=-infinity..infinity),t).
For example,

Download moment.mw

See ?Statistics[ProbabilityDensityFunction] for more details.

Edit. The definition of M[X](t).

Moment

acer — Thu, 08 Nov 2012 23:30:07 Z

You can use the Statistics:-Moment command.

You could do this starting with a random variable produced wih the RandomVariable command. But you mentioned getting there from the pdf, so let's start with that:

restart:
with(Statistics):

assume(sigma::real):
pdf:=(1/2)*2^(1/2)*exp(-(1/2)*(t-a)^2/sigma^4)/(Pi^(1/2)*sigma^2):

X:=Distribution(PDF=unapply( pdf, t )):

Moment(X,5);
                 5       3      4             8
                a  + 10 a  sigma  + 15 a sigma 

restart:
with(Statistics):

pdf:=(1/2)*2^(1/2)*exp(-(1/2)*(t-a)^2/sigma^4)/(Pi^(1/2)*sigma^2):

X:=Distribution(PDF=unapply( pdf, t )):

Moment(X,5):
simplify(%)  assuming sigma::real;

                 / 4       2      4           8\
               a \a  + 10 a  sigma  + 15 sigma /

acer

Addition

Markiyan Hirnyk — Fri, 09 Nov 2012 00:30:39 Z

Take a look at ?Statistics[MomentGeneratingFunction] .

Moment-generating function, not moments

Markiyan Hirnyk — Fri, 09 Nov 2012 00:09:44 Z

The question is how to find the moment-generating function of an absolutely continuous random variable, not its moments.

careless

acer — Fri, 09 Nov 2012 01:03:45 Z

@Markiyan Hirnyk Sorry, that was careless of me; I didn't read the question at all carefully.