Page 179 - DMTH404_STATISTICS

Unit 13: Moment Generating Function - Continue

Solution Notes

We can use the following covariance formula:
Cov[X , X ] = E[X X ] – E[X ]E[X ]
1 2 1 2 1 2
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 , (t ,0)
X 1 X 2 1
= E[exp(t X + 0.X )]
1 1 2
= M , (t , 0)
X X 2 1
1
1

= [1  exp(t  2.0) exp(2t  0)]
1
1
3
1
= [1  exp(t ) exp(2t )]

1
1
3
The expected value of X is obtained by taking the first derivative of its moment generating
1
function
dM (t ) 1
X 1

1  [exp(t ) 2exp(2t )]
dt 3 1 1
1
and evaluating it at t = 0
1
dM (t ) 1
E[X ] = X 1 1  [exp(0) 2exp(0)] 1


1
dt 3
1
1 t  0
The moment generating function of X is
2
M (t ) = E[exp(t X )]
X 2 2 2 2
= E[exp(0.X + t X )]
1 2 2
= M , (0,t )
X X 2
1 2
1




= [1 exp(0 2t ) exp(2.0 t )]
2
2
3
1

= [1 exp(2t ) exp(t )]

2
2
3
To compute the expected value of X we take the first derivative of its moment generating
2
function
dM (t )  1 [2exp(t ) exp(t )]
X2
2

dt 3 2 2
2
and evaluating it at r = 0
2
dM (t ) 1
E[X ]  X2 2  [2exp(0) exp(0)] 1


2
dt 3
2
2 t  0
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