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Moment Generating Functions
Lecture X
I.
Moment Generating Functions
A.
Definition
2.3.3.
Let
X
be
a
random
variable
with
cumulative
distribution function
FX
.
The moment generating function (mgf) of
X
(or
), denoted
X
Mt
, is
tX
X
M
t
E e
provided that the expectation exists for
t
in some neighborhood of 0.
That is, there is an
0
h
such that, for all
t
in
h
t
h
,
tX
Ee
exists.
1.
If the expectation does not exist in a neighborhood of 0, we say that the
moment generating function does not exist.
2.
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
X
, then
0
n
n
X
E X
M
where we define
()
0
0
n
n
XX
n
t
d
M
M
t
dt
1.
First note that
tX
e
can be approximated around zero using a Taylor series
expansion:
23
0
0
2
0
3
0
11
0
0
0
26
1
tx
t
t
t
X
M
t
E e
E e
te
x
t e
x
t e
x
tt
E x t
E x
E x
Note for any moment
n
:
1
2
2
n
n
n
n
n
n
d
M
M
t
E x
E x
t
E x
t
dt
Thus, as
0
t
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View Full DocumentAEB 6933 Mathematical Statistics for Food and Resource Economics
Lecture X
Professor Charles B. Moss
Fall 2007
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 Fall '09
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