Stat 5101 (Geyer) Fall 2009
Homework Assignment 10
Due Wednesday, December 2, 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.
101.
Suppose (
X
1
, X
2
) is a bivariate normal random vector, and assume
it is nondegenerate. Write its PDF in terms of the mean vector and variance
matrix. Then rewrite its PDF in terms of new parameters, which are
E
(
X
1
) =
μ
1
E
(
X
2
) =
μ
2
sd(
X
1
) =
σ
1
sd(
X
2
) =
σ
2
cov(
X
1
, X
2
) =
ρσ
1
σ
2
Then simplify your expression for the PDF so it contains no matrices, no
matrix inverses, determinants, or matrix multiplication.
Hint: the inverse of a matrix
A
=
a
11
a
12
a
21
a
22
can be done by Cramer’s rule obtaining
A

1
=
1
det(
A
)
a
22

a
12

a
21
a
11
assuming
A
is invertible, which it is if det(
A
) is not zero.
102.
Suppose (
X
1
, X
2
) is a nondegenerate bivariate normal random vec
tor.
Calculate the conditional PDF of
X
1
given
X
2
not using the theory
developed in class. Just use conditional = joint
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 Fall '02
 Staff
 Normal Distribution, Standard Deviation, Probability theory, ODOT, normal random vector

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