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Math 370, Actuarial Problemsolving
Momentgenerating functions
Momentgenerating functions
Deﬁnitions and properties
•
General deﬁnition of an mgf:
M
(
t
) =
M
X
(
t
) = E(
e
tX
).
•
Mgf of discrete r.v.’s:
M
(
t
) =
∑
e
tx
f
(
x
), where
f
(
x
) is the p.m.f. of
X
and the sum is taken
over all values
x
of
X
.
•
Mgf of continuous r.v.’s:
M
(
t
) =
R
*
*
e
tx
f
(
x
)
dx
, where
f
(
x
) is the p.d.f. of
X
, and the range of
integration is the range on which
f
(
x
) is deﬁned.
•
Expectation, variance, and moments via mgf’s:
M
(0) = 1,
M
0
(0) = E(
X
),
M
00
(0) = E(
X
2
),
M
000
(0) = E(
X
3
), etc.; Var(
X
) =
M
00
(0)

M
0
(0)
2
.
•
Mgf of a multiple of a r.v.:
If
X
has mgf
M
X
(
t
), and
Y
=
cX
with
c
a constant, then the mgf
of
Y
is
M
Y
(
t
) =
E
(
e
tY
) =
E
(
e
tcX
) =
M
X
(
tc
).
•
Mgf of a sum of independent r.v.’s
X
and
Y
:
If
X
and
Y
are independent, then
X
+
Y
has
mgf
M
X
+
Y
(
t
) =
M
X
(
t
)
M
Y
(
t
). (An analogous formula holds for sums of more than two independent
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 Spring '11
 rsharma

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