STAT 410 HW #3 Answers
Due Wednesday, October 15, 2003
You can turn it in to my office, 116B IH, or mailbox in 101 IH.
1
. The Gamma Distribution has two parameters, the shape
p >
0 and the scale
λ >
0, and
is denoted
Gamma
(
p, λ
). Its space is (0
,
∞
), and the pdf is
f
(
y
;
p, λ
) =
c y
p

1
e

λy
.
(a)
Show that
c
=
λ
p
Γ(
p
)
.
Answer:
1
c
=
∞
0
y
p

1
e

λy
dy
(set
u
=
λy
)
=
∞
0
(
u/λ
)
p

1
e

u
du/λ
=
1
λ
p
∞
0
u
p

1
e

u
du
=
1
λ
p
Γ(
p
)
.
(b)
χ
2
ν
and
Exponential
(
λ
) are special cases of the Gamma. What are the corresponding
(
p, λ
)’s?
Answer:
χ
2
ν
=
Gamma
(
ν
2
,
1
2
) and
Exponential
(
λ
) =
Gamma
(1
, λ
).
(c)
If
Y
∼
Gamma
(
p, λ
), then what is the distribution of
aY
for positive constant
a
?
Answer: Let
X
=
aY
, so that
y
=
x/a
and the Jacobian is 1
/a
. The the pdf of
X
is
f
X
(
x
) =
f
Y
(
x/a
)
/a
=
λ
p
Γ(
p
)
(
y/a
)
p

1
e

λy/a
/a
=
(
λ/a
)
p
Γ(
p
)
y
p

1
e

(
λ/a
)
y
,
which is
Gamma
(
p, λ/a
).
2
. Suppose
X
1
and
X
2
are independent, with
X
1
∼
Gamma
(
p
1
, λ
) and
X
2
∼
Gamma
(
p
2
, λ
).
Let
Y
1
=
X
1
+
X
2
and
Y
2
=
X
1
X
1
+
X
2
.
(a)
What is the space of (
Y
1
, Y
2
)?
1
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Answer:
Y
= (0
,
∞
)
×
(0
,
1).
(b)
Find
g

1
(
y
1
, y
2
) and the Jacobian.
Answer:
x
1
=
y
1
y
2
and
x
2
=
y
1

x
1
=
y
1
(1

y
2
), so
g

1
(
y
1
, y
2
) = (
y
1
y
2
, y
1
(1

y
2
)). The
Jacobian is then
∂y
1
y
2
∂y
1
∂y
1
(1

y
2
)
∂y
1
∂y
1
y
2
∂y
2
∂y
1
(1

y
2
)
∂y
2
=
y
2
1

y
2
y
1

y
1
=

y
2
y
1

y
1
(1

y
2
) =

y
1
.
(c)
Find the pdf of (
Y
1
, Y
2
).
Answer:
f
X
(
x
1
, x
2
) =
λ
p
1
Γ(
p
1
)
λ
p
2
Γ(
p
2
)
x
p
1

1
1
x
p
2

1
2
e

λ
(
x
1
+
x
2
)
,
so
f
Y
(
y
1
, y
2
)
=
f
X
(
y
1
y
2
, y
1
(1

y
2
))
 
y
1

=
λ
p
1
Γ(
p
1
)
λ
p
2
Γ(
p
2
)
(
y
1
y
2
)
p
1

1
(
y
1
(1

y
2
))
p
2

1
e

λy
1
y
1
=
λ
p
1
+
p
2
Γ(
p
1
)Γ(
p
2
)
y
p
1
+
p
2

1
1
e

λy
1
y
p
1

1
2
(1

y
2
)
p
2

1
(d)
Are
Y
1
and
Y
2
independent? What are their distributions? (They should be Gamma
and Beta, respectively.) How does
Y
2
depend on
λ
?
Answer: Yes, they are independent, because their joint pdf factors into a function of just
y
1
and a function of just
y
2
, and the space
Y
is a rectangle. By shifting around constants, we
can write
f
Y
(
y
1
, y
2
) =
λ
p
1
+
p
2
Γ(
p
1
+
p
2
)
y
p
1
+
p
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 Spring '08
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 Trigraph, Emoticon, p1, Conjugate prior, Dirichlet distribution

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