A random variable

has cdf of the form

,

is real,

is real and

.

Find the pdf function of

.

Find the form of the moment generating function

and hence calculate the variance of

[Note : the gamma function,

is such that

.

Letting

.

The pdf turns out to be

.

Using the fact that if

then

,

It turns out that

Using

and

We then have

which, according to Maple, is equal to

.

But then I end up with dividing by zero errors when trying to calculate

.

Plus I don't see how nor where to use the note given in the exercice.

Any help would be much appreciated. Thank you.