PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
Remark
The marginal distribution of X is sometimes called the negative binomial distribution.
In particular, when α is an integer, it corresponds to the definition previously with
p =
λ
1 + λ .
Examples
1. Suppose X ∼ Bernoulli with parameter p. Then M X (t) = 1 + p(e t − 1).
Now suppose X 1 , X 2 , . . . , X n are i.i.d. Bernoulli with parameter p and
Y = X 1 + X 2 + · · · + X n ;
then M Y (t) = ( 1 + p(e t − 1) ) n
, which agrees with the formula previously given
for the binomial distribution.
2. Suppose X 1 ∼ N(µ 1 , σ 2 1) and X 2 ∼ N(µ 2 , σ 2 2) independently. Find the MGF of
Y = X 1 + X 2 .
Solution. Recall that M X1 (t) = e tµ 1
e t2 σ 2 1 /2
M X2 (t) = e tµ 2
e t2 σ 2 2 /2
⇒ M Y (t) = M X1 (t)M X2 (t)
= e t(µ 1+µ 2 ) e t2 (σ 2 1 +σ2 2 )/2
⇒ Y ∼ N ( µ 1 + µ 2 , σ 2 1 + σ 2 2)
.
1.9.2 Marginal distributions and the MGF
To find the moment generating function of the marginal distribution of any set of
components of X, set to 0 the complementary elements of t in M X (t).
Let X = (X 1 , X 2 ) T , and t = (t 1 , t 2 ) T . Then
( )
t1
M X1 (t 1 ) = M X .
0
To see this result: Note that if A is a constant matrix, and b is a constant vector, then
M AX+b (t) = e tT b M X (A T t).
53