Page 171 - DMTH404_STATISTICS

Unit 12: The Moment Generating Functions

Notes
Example 2: Let X be a random variable with moment generatinf function

1
M (t) = 1 exp(t) 
2
X
Derive the variance of X.
Solution
We can use the following formula for computing the variance

Var[X] = E[X ] – E[X] 2
2
The expected value of X is computed by taking the first derivative of the moment generating
function.

dM (t)  1 exp(t)
X
dt 2
and evaluating it at t = 0

dM (t) 1 1
E[X]  X  exp(0) 
dt t 0 2 2

The second moment of X is computed by taking the second derivative of the moment generating
function
2
d M (t)  1 exp(t)
X
dt 2 2
and evaluating it at t = 0
2
d M (t) 1 1
2
E[X ]  X  exp(0) 
dt 2 2 2
t 0

2
Var[X] = E[X ] – E[X] 2
2
1  1 
=   
2  2 
1 1
= 
2 4
1
=
4

Example 3: A random variable X is said to have a Chi-square distribution with n degrees
1
of freedom if its moment generating function is defined for any t  and it is equal to:
2
M (t) = (1 – 2t) -n/2
X

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