Stat 5101 (Geyer) Fall 2011
Homework Assignment 2
Due Wednesday, September 21, 2011
Solve each problem. Explain your reasoning. No credit for answers with
no explanation.
21.
Suppose we have a PMF
f
X
with domain
S
(the original sample
space), and we have a map
g
:
S
→
T
that induces a probability model with
PMF
f
Y
with domain
T
(the new sample space) given by the formula on
slide 82. Prove that for any realvalued function
h
on
S
X
y
∈
T
h
(
y
)
f
Y
(
y
) =
X
x
∈
S
h
(
g
(
x
)
)
f
X
(
x
)
22.
Suppose
f
X
is the uniform distribution on
S
=
{
2
,

1
,
0
,
1
,
2
}
and
the random variable
X
defined by
X
(
s
) =
s
2
,
s
∈
S.
Determine the PMF of the random variable
X
.
23.
Suppose
X
= (
X
1
, X
2
) has the uniform distribution on
{
1
,
2
,
3
,
4
,
5
,
6
}
2
.
(a) Show that the components of
X
are independent.
(b) Determine the PMF of the distribution of the random variable
Y
=
X
1
+
X
2
.
(c) Determine
E
(
Y
).
(d) Define
μ
=
E
(
Y
). Determine
E
{
(
Y

μ
)
2
}
.
24.
Suppose
X
= (
X
1
, X
2
, X
3
) has the uniform distribution on the set
{
(0
,
0
,
0)
,
(1
,
1
,
0)
,
(1
,
0
,
1)
,
(0
,
1
,
1)
}
Show that the components of
X
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 Spring '11
 Staff
 Probability theory, large number, different poker hands

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