PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
Proof. (Theorem 1.9.6, continuous case)
M X (t 1 , . . . , t r ) =
=
∫ ∞
−∞
∫ ∞
−∞
e t 1x 1 +t 2 x 2 +···+t rx r
f X (x 1 , . . . , x r ) dx 1 . . . dx r
∫ ∞
. . . e t 1x 1
e t 2x 2
. . . e trxr f X1 (x 1 )f X2 (x 2 ) . . . f Xr (x r ) dx 1 . . . dx r
−∞
(∫ ∞
) (∫ ∞
) (∫ ∞
)
= e t 1x 1
f X1 (x 1 ) dx 1 e t 2x 2
f X2 (x 2 ) dx 2 . . . e trxr f Xr (x r ) dx r
−∞
−∞
−∞
= M X1 (t 1 )M X2 (t 2 ) . . . M Xr (t r ), as required.
We saw previously that if Y = h(X), then we can find M Y (t) = E[e th(X) ] from the
joint distribution of X 1 , . . . , X r without calculating f Y (y) explicitly.
A simple, but important case is:
Theorem. 1.9.7
Y = X 1 + X 2 + · · · + X r .
Suppose X 1 , . . . , X r are RVs and let Y = X 1 + X 2 + · · · + X r . Then (assuming MGFs
exist):
1. M Y (t) = M X (t, t, t, . . . , t).
2. If X 1 . . . X r are independent then M Y (t) = M X1 (t)M X2 (t) . . . M Xr (t).
3. If X 1 , . . . , X r independent and identically distributed with common MGF M X (t),
then:
M Y (t) = [ M X (t) ] r
.
51