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Let X be a random variable for which the momentgenerating function exists.
Let
denote its
mean.
For j = 2, 3, .
.. , let
j
=
E[(X – μ)
j
]
be the jth central moment of X (hence
2
is the variance).
We can expand the function
ln(E[e
tX
]), called the
cumulant generating function
of X, as a power series in t as follows.
ln(E[e
tX
]) =
ln(E[e
t
+ t(X–
)
])
=
μt
+
ln(E[e
t(X–
)
])
=
μt
+
ln(E[
])
because e
z
=
n
z
n!
=
μt
+
ln(1
+
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 Spring '08
 Tang,Q

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