Stat 5101 (Geyer) Fall 2011
Homework Assignment 3
Due Wednesday, September 28, 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.
31.
Suppose that
f
is a PMF on a sample space
S
, suppose
X
and
Y
are
random variables in this probability model. Prove the following statements.
(a)
E
(
X
+
Y
) =
E
(
X
) +
E
(
Y
).
(b) If
X
(
s
)
≥
0 for all
s
∈
S
, then
E
(
X
)
≥
0.
(c) If
Y
(
s
) =
a
for all
s
∈
S
, then
E
(
XY
) =
aE
(
X
).
(d) If
Y
(
s
) = 1 for all
s
∈
S
, then
E
(
Y
) = 1.
Do not use the axioms (these are the axioms). The problem is to prove that
these statements follow from our earlier deﬁnition of PMF and expectation.
32.
Suppose
X
has the uniform distribution on the set
{
1
,
2
,
3
,
4
}
, and
suppose
Y
=
X
2
.
(a) Calculate
E
(
X
).
(b) Calculate
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 Spring '11
 Staff
 Probability theory, Calculate

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