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University of California, Los Angeles
Department of Statistics
Statistics 100B
Instructor: Nicolas Christou
Moment generating functions
Deﬁnition:
M
X
(
t
) =
Ee
tX
Therefore,
If
X
is discrete
M
X
(
t
) =
X
x
e
tX
P
(
x
)
If
X
is continuous
M
X
(
t
) =
Z
x
e
tX
f
(
x
)
dx
Aside:
e
x
= 1 +
x
1!
+
x
2
2!
+
x
3
3!
+
···
Similarly,
e
tx
= 1 +
tx
1!
+
(
tx
)
2
2!
+
(
tx
)
3
3!
+
···
Let
X
be a discrete random variable.
M
X
(
t
) =
X
x
e
tX
P
(
x
) =
X
x
"
1 +
tx
1!
+
(
tx
)
2
2!
+
(
tx
)
3
3!
+
···
#
P
(
x
)
or
M
X
(
t
) =
X
x
P
(
x
) +
t
1!
X
x
xP
(
x
) +
t
2
2!
X
x
x
2
P
(
x
) +
t
3
3!
X
x
x
3
P
(
x
) +
···
To ﬁnd the
k
th
moment simply evaluate the
k
th
derivative of the
M
X
(
t
) at
t
= 0.
EX
k
= [
M
X
(
t
)]
k
th
derivative
t
=0
For example:
First moment:
M
X
(
t
)
0
=
X
x
xP
(
x
) +
2
t
2!
X
x
x
2
P
(
x
) +
···
We see that
M
X
(0)
0
=
∑
x
xP
(
x
) =
E
(
X
).
1
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View Full Document Similarly,
Second moment
M
X
(
t
)
00
=
X
x
x
2
P
(
x
) +
6
t
3!
X
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This note was uploaded on 01/14/2011 for the course STATS 100B 100B taught by Professor Christou during the Winter '10 term at UCLA.
 Winter '10
 Christou

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