I know you said "without making any assumptions," but this one is pretty important. Assuming you mean are shape/rate parameters (as opposed to shape/scale), the PDF of
is
if , and 0 otherwise.
The MGF of is given by
Note that the integral converges only when .
Define
Integrate by parts, with
so that
Note that
By substitution, we have
and so on, down to
The integral of interest then evaluates to
so the MGF is
The first moment/expectation is given by the first derivative of at
.
Variance is defined by
The second moment is given by the second derivative of the MGF at .
Then the variance is
Note that the power series expansion of the MGF is rather easy to find. Its Maclaurin series is
where is the
-derivative of the MGF evaluated at
. This is also the
-th moment of
.
Recall that for ,
By differentiating both sides 7 times, we get
Then the -th moment of
is
and we obtain the same results as before,
and the same variance follows.
