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Chapter 03  Random Variables
CHAPTER 3
RANDOM VARIABLES
3.1.a.
∑
)
(
P
x
= 1.0
b.
x
F(
x
)
0
0.3
1
0.5
2
0.7
3
0.8
4
0.9
5
1.0
c. P(
X
> 2) = 1

F(2) = 0.3
3.2.a.
∑
)
(
P
x
= 1.0
b.
x
F(
x
)
0
0.01
1
0.10
2
0.40
3
0.60
4
0.80
5
0.90
6
1.00
c.
)
4
(
P
≤
X
= F(4) = 0.80
d.
)
2
(
P
≥
X
= 1

F(1) = 0.90
3.3.a.
∑
)
(
P
x
= 1.0
b.
x
F(
x
)
0
0.10
10
0.30
20
0.65
30
0.85
40
0.95
50
1.00
c. P(
X
> 20) = 1

F(20) = 0.35
31
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3.4.a.
)
4
2
(
P
≤
≤
X
= 0.2 + 0.2 + 0.3 = 0.7
b.
x
F(
x
)
0
0.1
1
0.2
2
0.4
3
0.6
4
0.9
5
1.0
c.
∑
)
(
P
x
= 1.0
3.5.first die outcome:
1
2
3
4
5
6
1
•
•
•
•
•
•
2
•
•
•
•
•
•
3
•
•
•
•
•
•
Second
4
•
•
•
•
•
•
die out
5
•
•
•
•
•
•
come:
6
•
•
•
•
•
•
X
= sum of the two dice. Note from the sample space that one outcome leads to a sum of 2, two equally
likely outcomes lead to a sum of 3, three lead to a sum of 4, four to a sum of 5, five to a sum of 6, and
six (the highest number of outcomes) lead to a sum of 7. Thus, the most likely sum is
x
= 7. Afterwards,
the probabilities decline: five outcomes lead to a sum of 8, four to a sum of 9, three to a sum of 10, two
to a sum of 11, and one outcome (two “sixes”) leads to a sum of 12. We thus have the following
probability distribution and cumulative distribution function:
X
P(
x
)
F(
x
)
2
1/36
1/36
3
2/36
3/36
4
3/36
6/36
5
4/36
10/36
6
5/36
15/36
7
6/36
21/36
8
5/36
26/36
9
4/36
30/36
10
3/36
33/36
11
2/36
35/36
12
1/36
1.0
32
Chapter 03  Random Variables
36.
a.
∑
)
(
P
x
= 1.0
b.
x
F(
x
)
0
0.1
1
0.3
2
0.7
3
0.8
4
0.9
5
1.00
c.
)
4
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This note was uploaded on 11/24/2010 for the course MATH BAOI taught by Professor Phong during the Spring '10 term at Wash. College.
 Spring '10
 Phong

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