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STATS 116
Kihwan Choi
Homework 4 solutions
1. (3.3.9) Out of
n
individual voters at an election,
r
vote Republican and
n

r
vote Democrat. At the next
election the probability of a Republican switching to vote Democrat is
p
1
, and of a Democrat switching is
p
2
.
Suppose individuals behave independently. Find (a) the expectation and (b) the variance of the number of
Republican votes at the second election.
Solution:
(a) Let
X
denote the number of Republican votes at the second election. Also
X
1
and
X
2
denote the number
of Republican votes at the second election with Republican votes at the ﬁrst election and the number
of Republican votes at the second election with Democrat votes at the ﬁrst election, respectively. Then
X
=
X
1
+
X
2
where
X
1
∼
B
(
r,
1

p
1
) and
X
2
∼
B
(
n

r,p
2
). Thus,
EX
=
EX
1
+
EX
2
=
r
(1

p
1
) + (
n

r
)
p
2
(b) Since
X
1
and
X
2
are independent,
V ar
(
X
) =
V ar
(
X
1
) +
V ar
(
X
2
) =
rp
1
q
1
+ (
n

r
)
p
2
q
2
2. (3.3.12) A random variable
X
has expectation 10 and standard deviation 5.
(a) Find the smallest upper bound you can for
P
(
X
≥
20).
Solution:
By onesided Chebyshev’s inequality (Cantelli’s inequality),
P
(
X
≥
20) =
P
(
X

μ
σ
≥
2)
≤
1
1 + 2
2
= 0
.
2
,
which can be achievable when
P
(
X
) =
‰
0
.
2 for
X
= 20
,
0
.
8 for
X
= 7.5
,
resulting in
μ
= 10 and
σ
= 5.
(b) Could
X
be a binomial random variable?
Solution:
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 Spring '08
 ROSS
 Probability

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