SIEO 3600 (IEOR Majors)
Solution to Assignment #8
Introduction to Probability and Statistics
March 31, 2010
Solution to Assignment #8
(
Binomial, Geometric and Uniform random variables
)
1.
Polling:
During a Presidential election (only two candidates for simplicity).
(a) It has been determined (by extensive sample) that
p
= 55% of all voters will vote for
candidate
A
and
q
= 1

p
= 45% for candidate
B
.
If you randomly select 5 voters,
what is the probability that exactly 3 will vote for candidate
A
?
(b) Suppose that apriori we do not know the proportion
p
of voters who will vote for can
didate
A
, but that when we randomly selected 5 voters, exactly 3 said they would vote
for
A
. Find the value of
p
that maximizes the probability of this event (e.g., of getting
exactly 3 out the 5 saying they would vote for
A
.)
Solution:
(a) Let
X
be the number of voters who vote for
A
. Then
X
is a binomial random variable
with parameter
p
= 0
.
55 and
n
= 5. Therefore,
P
(
X
= 3) =
5
3
0
.
55
3
0
.
45
2
.
(b) We know that
X
is a geometric random variable with
n
= 5 but unknown
p
. Also, we
have
P
(
X
= 3) =
5
3
p
3
(1

p
)
2
.
(1)
We want to maximize
P
(
X
= 3) such that 0
< p <
1. Taking the derivative of (1) with
respect to
p
and let it be 0, we have
d
dp
P
(
X
= 3) =
5
3
(
3
p
2
(1

p
)
2
+
p
3
2(1

p
)(

1)
)
= 0
,
(2)
implies that
p
*
= 3
/
5 is the maximizer.
2. Let
X
be a binomial random variable with
E
[
X
] = 7
and
var(
X
) = 2
.
1
Find
(a)
P
(
X
= 4);
(b)
P
(
X >
12).
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SIEO 3600, Solution to Assignment #8
Solutions:
(a) Because
X
follows a binomial
Bino
(
n, p
), therefore
E
[
X
] =
np
= 7
var(
X
) =
np
(1

p
) = 2
.
1
⇒
n
= 10
p
= 0
.
7
Hence,
P
(
X
= 4) =
(
10
4
)
0
.
7
4
0
.
3
6
.
(b) Since
n
= 10, the state space of
X
is
S
X
=
{
0
,
1
,
2
, . . . ,
10
}
, so
P
(
X >
12) = 0.
3. Suppose that a particular trait (such as eye color or lefthandedness) of a person is classified
on the basis of one pair of genes, and suppose that
d
represents a dominant gene and
r
a
recessive gene. Thus, a person with
dd
genes is pure dominance, one with
rr
is pure recessive,
and one with
rd
is hybrid.
The pure dominance and the hybrid are alike in appearance.
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 Spring '10
 YUNANLIU
 Probability, Probability theory, Cumulative distribution function, Discrete probability distribution, binomial random variable

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