Stat 5101 (Geyer) Fall 2011
Homework Assignment 7
Due Wednesday, November 2, 2011
Solve each problem. Explain your reasoning. No credit for answers with
no explanation. If the problem is a proof, then you need words as well as
formulas. Explain why your formulas follow one from another.
71.
If
X
has the Gam(
α,λ
) distribution, we calculated in class that
E
(
X
β
) =
Γ(
α
+
β
)
Γ(
α
)
λ
β
.
(a) Find
E
(
X
2
)
(b) Find var(
X
).
None of your answers should contain gamma functions (use the gamma
function recursion formula to simplify).
72.
If
X
has the Beta(
α
1
,α
2
) distribution, show that
E
{
X
β
1
(1

X
)
β
2
}
=
Γ(
α
1
+
α
2
)Γ(
α
1
+
β
1
)Γ(
α
2
+
β
2
)
Γ(
α
1
)Γ(
α
2
)Γ(
α
1
+
α
2
+
β
1
+
β
2
)
Hint: use the fact that the PDF of the beta distribution integrates to one,
just like we did for the gamma distribution. You may ignore the issue of
when the integral exists (it exists when
β
1
>

α
1
and
β
2
>

α
2
, but we
don’t know how to prove that yet).
73.
Suppose
X
has the Beta(
α
1
,α
2
) distribution.
(a) Find
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 Spring '11
 Staff
 Normal Distribution, Probability theory

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