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Grinstead-Snell-Probability
· Problem 10.3.3
Markov Chains
Mathematics · moment-generating-functions-continuous
Problem
Let
X
X
X
be a continuous random variable with values in
[
0
,
∞
)
[0, \infty)
[
0
,
∞
)
and density
f
X
f_X
f
X
. Find the moment generating functions for
X
X
X
if (a)
f
X
(
x
)
=
2
e
−
2
x
f_X(x) = 2e^{-2x}
f
X
(
x
)
=
2
e
−
2
x
. (b)
f
X
(
x
)
=
e
−
2
x
+
(
1
/
2
)
e
−
x
f_X(x) = e^{-2x} + (1/2)e^{-x}
f
X
(
x
)
=
e
−
2
x
+
(
1/2
)
e
−
x
. (c)
f
X
(
x
)
=
4
x
e
−
2
x
f_X(x) = 4xe^{-2x}
f
X
(
x
)
=
4
x
e
−
2
x
. (d)
f
X
(
x
)
=
λ
(
λ
x
)
n
−
1
e
−
λ
x
/
(
n
−
1
)
!
f_X(x) = \lambda(\lambda x)^{n-1} e^{-\lambda x}/(n-1)!
f
X
(
x
)
=
λ
(
λ
x
)
n
−
1
e
−
λ
x
/
(
n
−
1
)!
.
Related theory
Characteristic Functions And Moment Generating Functions
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