Stat 5101 (Geyer) Fall 2009
Homework Assignment 7
Due Friday, November 6, 2009
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.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '02
 Staff
 Normal Distribution, Probability theory

Click to edit the document details