Statistics 5101, Fall 2010: Homework
Assignment 3
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 pr is a PMF on a sample space Ω, 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
(
ω
)
≥
0 for all
ω
∈
Ω, then
E
(
X
)
≥
0.
(c) If
Y
(
ω
) =
a
for all
ω
∈
Ω, then
E
(
XY
) =
aE
(
X
).
(d) If
Y
(
ω
) = 1 for all
ω
∈
Ω, 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 sup
pose
Y
=
X
2
.
(a) Calculate
E
(
X
).
(b) Calculate
E
(
Y
).
(c) Calculate
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 Fall '02
 Staff
 Statistics, Probability theory, mean vector, Calculate, Calculate Var, variance matrix

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