Page 169 - DMTH404_STATISTICS

Unit 12: The Moment Generating Functions

Notes
dM (t)  1
E[X]  X  
dt t 0 (  0) 2 

The second moment of X can be computed by taking the second derivative of the moment
generating function

2
d M (t)  2
X
dt 2 (  t) 3
and evaluating it at r = 0

2
d M (t) 2 2
E[X ] = X  
2 2 3 2
dt (  t) 

t 0
And so on for the higher moments.
12.3 Characterization of a Distriution via the mgf

The most important property of the moment generating function is the following:
Proposition (Equality of distributions) Let X and Y be two random variables. Denote by F (x)
X
and F (y) their distribution functions and by M (t) and M (t) their moment generating functions.
Y X Y
X and Y have the same distribution (i.e., F (x) = F (x) for any x) if and only if they have the same
X Y
moment generating functions (i.e. M (t) = M (t) for any t).
X Y
While proving this proposition is beyond the scope of this introductory exposition, it must be
stressed that this proposition is extremely important and relevant from a practical viewpoint in
many cases where we need to prove that two distributions are equal, it is much easier to prove
equality of the moment generating functions than to prove equality of the distribution functions.
Also note that equality of the distribution functions can be replaced in the proposition above by
equality of the probability mass function (if X and Y are discrete random variables) or by
equality of the probability density functions (if X and Y are absolutely continuous random
variables).

12.4 Moment Generating Function - More details

12.4.1 Moment Generating Function of a Linear Transformation

Let X be a random variable possessing a moment generating function M (t). Define:
X
Y = a + bX
where a, b   are two constants and b  0. Then, the random variable Y possesses a moment
generating function M (t) and
Y
Proof
Using the definition of moment generation function
MY(t) = E[exp(tY)]

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

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