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Grinstead-Snell-Probability
· Problem 10.1.11
Generating Functions
Mathematics · generating-functions-discrete
Problem
Show that if
X
X
X
is a random variable with mean
μ
\mu
μ
and variance
σ
2
\sigma^2
σ
2
, and if
X
∗
=
(
X
−
μ
)
/
σ
X^* = (X - \mu)/\sigma
X
∗
=
(
X
−
μ
)
/
σ
is the standardized version of
X
X
X
, then
g
X
∗
(
t
)
=
e
−
μ
t
/
σ
g
X
(
t
σ
)
.
g_{X^*}(t) = e^{-\mu t/\sigma} g_X\left(\frac{t}{\sigma}\right).
g
X
∗
(
t
)
=
e
−
μ
t
/
σ
g
X
(
σ
t
)
.
Related theory
Characteristic Functions And Moment Generating Functions
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