Page 170 - DMTH404_STATISTICS

Statistics

Notes = exp(at)E[exp(btX)]

= exp(at)M (bt)
X
Obviously, if M (t) is defined on a closed interval [–h, h], then M (t) is defined on the interval
X Y
  h h  .
,
 
 b b 
12.4.2 Moment Generating Function of a Sum of Mutually Independent
Random Variable

Let X , ..., X be n mutually independent random variables. Let Z be their sum
1 n
n
Z   X i

i 1
Then, the moment generating function of Z is the product of the moment generating functions
of X , ..., X .
1 n
n
M (t)   M X (t)
Z
Y

i 1
This is easily proved using the definition of moment generating function and the properties of
mutually independent variables (mutual independence via expectations):
MZ(t) = E[exp(tZ)]

  n  
 
= E exp t X l  

  i 1 

  n  
= E exp  tX l  

  i 1 

 n 
= E  exp(tX )
i 
 i 1 

n
=   E exp(tX )  (by mutual independence)
i

i 1
n
=  M (t) (by the definition generation function)
X
i

i 1
Example 1: Let X be a discrete random variable having a Bernoulli distribution. Its
support R is
X
R = <0, 1>
X
and its probability mass function p (x) is
X
 p if x  1


p (x)   1 p if x  0
X


 0 if x R X
where p  (0, 1) is a constant. Derive the moment generating function of X, if it exists.
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