Page 178 - DMTH404_STATISTICS

Statistics

Notes

Notes The moment generating function M (t ) of a standard normal random variable
X i i
is defined for any t  . As a consequence, the joint moment generating function of X is
i
K
defined for any t   .

Example 2: Let
X = [X X ] T
1 2
be a 2 × 1 random vector with joint moment generating function

1 2
M , (t ,t )   exp(t  2t )
1
2
1
2
X
1 X
3 3
2
Derive the expected value of X .
1
Solution
The moment generating function of X is:
1
M (t ) = E[exp(t X )]
X 1 1 1 1
= E[exp(t X + 0.X )]
1 1 2
= M ,X (t ,0)
X 1 2 1
1 2
=  exp(t  2.0)
1
3 3
1 2
=  exp(t )
1
3 3
The expected value of X is obtained by taking the first derivative of its moment generating
1
function:
dM (t ) 2
X 1 1  exp(t )
dt 3 1
1
and evaluating it at t = 0:
1
dM (t ) 2 2
E[X ]  X 1 1  exp(0) 
1
dt 3 3
1
1 t  0
Example 3: Let
X = [X X ] T
1 2
be a 2 × 1 random vector with joint moment generating function
1

M , (t ,t )  [1  exp(t  2t ) exp(2t  t )]
X 1 X 2 1 2 3 1 2 1 2
Derive the covariance between X and X .
1 2

170 LOVELY PROFESSIONAL UNIVERSITY