Download If X and Y are independent continuous random variables then the

If X and Y are independent continuous random variables then the moment
generating function (mgf ) for Z = X + Y is the product of the separate moment
generating functions.
∫ ∫
tZ
ψZ (t) = E(e ) = E(e
t(X+Y )
)=
etx .ety f (x, y)dydx
where f (x, y) is the joint probability density of X and Y . If X and Y are independent
then the joint density is the product of the marginal densities, i.e.
f (x, y) = fX (x)fY (y)
In that case,
∫ ∫
∫ ∫
tx ty
∫
tx ty
e e f (x, y)dydx =
e e fX (x)fY (y)dydx =
∫
∫
tx
= e fX (x)dx. ety fY (y)dy = ψX (t)ψY (t).
tx
e fX (x)
(
∫
)
ety fY (y)dy dx
Note that this only works if X and Y are independent. A common application is the case
of a random sample of size n where we are often interested in the sum of n independent
identically distributed random variables, in which case the mgf is (ψX (t))n .