Devore 5
th
Edition, Section 3.2 #12 (pp. 108)
Let
X
= the number of tires on a randomly selected automobile
that are under inflated
a.)
Which of the following three
p
(
x
) functions is a legitimate pmf for
X
?
Why are the other two not allowed?
x
0
1
2
3
4
p(x)
0.3
0.2
0.1
0.05
0.05
p(x)
0.4
0.1
0.1
0.1
0.3
p(x)
0.4
0.1
0.2
0.1
0.3
Only the second p(x) satisfies the following two conditions for a legitimate pmf:
1.
( )
0
≥
x
p
2.
( )
1
4
0
=
∑
=
x
x
p
b.)
For the legitimate pmf of part (a), compute:
P(2
≤
X
≤
4) = p(x) + p(3) + p(4) = 0.1 + 0.1 + 0.3 = 0.5
P(
X
≤
2) = 0.4 + 0.1 + 0.1 = 0.6
P(
X
≠
0) = 1-0.4 = 0.6
c.)
Suppose:
p
(
x
) =
c
⋅
(5-
x
)
for
x
= 0, 1, …, 4
What is the value of
c
?
[
Hint
:
Σ
p
(
x
) = 1]
( )
∑
=
4
0
x
x
p
=
(
)
(
)
(
)
(
)
(
)
4
5
3
5
2
5
1
5
0
5
−
⋅
+
−
⋅
+
−
⋅
+
−
⋅
+
−
⋅
c
c
c
c
c
= 5c + 4c + 3c + 2c + 1c = 15 c
Because
= 1, we have c = 1/15
( )
∑
=
4
0
x
x
p
1

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