Page 167 - DMTH404_STATISTICS

Unit 12: The Moment Generating Functions

12.1 Moment Generating Function of a Random Variable Notes

Moment Generating Function - Defintion

We start this lecture by giving a definition of moment generating function.
Definition: Let X be a random variable. If the expected value:
E[exp(tX)]
exists and is finite for all real numbers t belonging to a closed interval [–h, h]  , with h > 0,
then we say that X possesses a moment generating function and the function M : [–h, h]  
X
defined by:
Mx(t) = E[exp(t(X)]
is called the moment generating function (or mgf) of X.

Moment Generating Function - Example

The following example shows how the moment generating function of an exponential random
variable is calculated.

Example: Let X be an exponential random variable with parameter   ...its supposed
Rx is the set of positive real numbers:
RX = [0, )
and its probability density function f (x) is:
x


 l exp( lx) if x Rx
fx(x)  

 0 if x Rx
Its moment generating function is computed as follows:

E[exp(tX)] =   exp(tx)fx(x)dx



= 0  exp(tx) exp( x)dx

=  0  exp((t   )x)dx

 1  
=  exp((t   )dx
 
 t   0 
 1 
=  0 
 
 t   

=
t  

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