Download Moment Generating Functions: Definition, Properties, and Applications and more Study notes Introduction to Macroeconomics in PDF only on Docsity!
Moment Generating Functions
Lecture X
I. Moment Generating Functions
A. Definition 2.3.3. Let X be a random variable with cumulative
distribution function F X. The moment generating function (mgf) of
X (or F X ), denoted M X t , is
tX M (^) X t E e
provided that the expectation exists for t in some neighborhood of 0.
That is, there is an h^0 such that, for all tin h^ t^ h,
tX
E e exists.
- If the expectation does not exist in a neighborhood of 0, we say that the
moment generating function does not exist.
- More explicitly, the moment generating function can be defined as:
for continuous random variables, and
for discrete random variables
tx X
tx X x
M t e f x dx
M t e P X x
B. Theorem 2.3.2: If X has mgf M^ X t , then
n n E X MX
where we define
( )
0
n n X (^) n X t
d M M t dt
- First note that
tX e can be approximated around zero using a Taylor series
expansion:
0 0 2 0 2 3 0 3
2 3 2 3
tx t t t X M t E e E e te x t e x t e x
t t E x t E x E x
Note for any moment n:
1 2 2
n n (^) n n n X (^) n X
d M M t E x E x t E x t dt
Thus, ast 0
Professor Charles B. Moss Fall 2007
n n M (^) X E x
- Leibnitz’s Rule: If f x, , a , and b are differentiable with respect
to , then
b
a
b
a
d d d f x dx f b a f a b d d d
f x dx
- Berger and Casella proof: Assume that we can differentiate under the
integral using Leibnitz’s rule, we have
tx X
tx
tx
d d M t e f x dx dt dt
d e f x dx dt
xe f x dx
Letting t 0 , this integral simply becomes
xf x dx E x
- This proof can be extended for any moment of the distribution function.
C. Moment Generating Functions for Specific Distributions
- Application to the Uniform Distribution:
tx bt at b b tx X a a
e e e M t dx e b a b a t t b a
Following the expansion developed earlier, we have:
2 2 2 3 3 3
2 2 2 3 3 3
2 2 2 3
2 2 2
X
b a t b a t b a t
M t b a t
b a t b a t
b a t b a t
b a b a (^) t b^ a^ b^ ab^ a t
b a t b a t
a b t a ab b t
Letting b 1 and a 0 , the last expression becomes:
Professor Charles B. Moss Fall 2007
(^2 2 2 4 )
2 2
2 2 2 2
x^ x^ t^ t^ t tx
x t t t
The moment generating function then becomes:
2 2 2 2
2 2
exp exp 2 2 2
exp 2
X
x t M t t t dx
t t
Taking the first derivative with respect to t, we get:
exp 2
M (^) X t t t t
Letting t^0 , this becomes: 1 MX 0
The second derivative of the moment generating function with respect tot
yields:
(^2 2 2 )
2 2 2 2
exp 2
exp 2
M (^) X t t t
t t t t
Again, letting t 0 yields (^2 2 ) MX 0
- Let X and Y be independent random variables with moment generating
functions M (^) X t and M (^) Y t. Consider their sum Z X Yand its
moment generating function:
tz t x^ y tx ty Z
tx ty X Y
M t E e E e E e e
E e E e M t M t
a) We conclude that the moment generating function for two independent random variables is equal to the product of the moment
generating functions of each variable.
b) Skipping ahead slightly, the multivariate normal distribution
function can be written as:
Professor Charles B. Moss Fall 2007
exp ' 2 2
f x x x
where is the variance matrix and is a vector of means.
c) In order to derive the moment generating function, we now need a
vector t. The moment generating function can then be defined as:
1 exp ' ' 2
X M (^) t t t t
d) Normal variables are independent if the variance matrix is a diagonal
matrix.
e) Note that if the variance matrix is diagonal, the moment generating
function for the normal can be written as:
1 2 3
2 1 2 2 2 3
2 2 2 2 2 2 1 1 2 2 3 3 1 1 2 2 3 3
2 2 2 2 1 1 1 1 2 2 2 3 3 3
exp ' ' 0 0 2 0 0
exp 2
exp 2 2 2
X
X X X
M t t t t
t t t t t t
t t t t
M t M t M t
