STAT 410
Examples for 10/15/2008
Fall 2008
Gamma Distribution
:
(
)
(
)
x
e
x
x
f
°
1
±
±
±
°


Γ
=
,
0
≤
x
<
∞
OR
(
)
(
)
²
1
±
1
±
²
±
x
e
x
x
f


Γ
=
,
0
≤
x
<
∞
If T has a Gamma
(
α
,
θ
=
1
/
λ
)
distribution, where
α
is an integer, then
F
T
(
t
)
= P
(
T
≤
t
)
= P
(
X
t
≥
α
)
,
where X
t
has a Poisson
(
λ
t
)
distribution.
1.
a)
Let X be a random variable with a chisquare distribution with
r
degrees of
freedom. Show that X has a Gamma distribution. What are
α
and
θ
?
M
X
(
t
)
= M
χ
2
(
r
)
(
t
)
=
(
)
2
2
1
1
r
t

,
t
< 2.
M
Gamma
(
α
,
θ
)
(
t
)
=
(
)
±
1
1
²
t

,
t
<
θ
.
If X has a chisquare distribution with
r
degrees of freedom, then X has a
Gamma distribution with
α
=
r
/
2
and
θ
= 2.
b)
Let Y be a random variable with a Gamma distribution with parameters
α
and
θ
=
1
/
λ
. Assume
α
is an integer. Show that
2
Y
/
θ
has a chisquare distribution.
What is the number of degrees of freedom?
M
Y
(
t
)
= M
Gamma
(
α
,
θ
)
(
t
)
=
(
)
±
1
1
²
t

,
t
<
θ
.
If W =
a
Y +
b
, then M
W
(
t
)
=
e
b
t
M
Y
(
a
t
)
.
M
2
Y
/
θ
(
t
)
= M
Y
(
2
t
/
θ
)
=
(
)
±
2
1
1
t

.
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 Spring '08
 AlexeiStepanov
 Normal Distribution, Probability, Standard Deviation, Variance, Probability theory, GAMMA

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