Assignment 5, EE 131A
Due: Monday November 20, 2006
Problem 1.
(a) First prove that the memoryless property of a random variable
X
P
[
X > k
+
j
v
v
X > j
] =
P
[
X > k
]
,
k, j >
0
,
is equivalent to the following relation:
P
[
X > k
+
j
] =
P
[
X > k
]
P
[
X > j
]
.
Prove that the geometric random variable satisfies this property.
(b) Suppose that
X
is a discrete random variable that satisfies the memoryless property. Show
that this implies that
X
is a geometric random variable. (
Hint:
Let
p
=
P
[
X
= 1]
and
q
= 1

p
.
By induction, show that
P
[
X
≤
k
] = 1

q
k
, and consequently show that
P
[
X
=
k
] =
q
k

1
p
.)
Problem 2.
The life of a certain type of automobile tire is normally distributed with mean
m
=
34
,
000
miles and standard deviation
σ
= 4000
miles.
(a) What is the probability that such a tire lasts over 40,000 miles?
(b) What is the probability that it lasts between 30,000 and 35,000 miles?
(c) Given that it has survived 30,000 miles, what is the conditional probability that it survives
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 Spring '08
 LORENZELLI
 Probability theory, 10,000 miles, 000 miles, 0.617 inches, 0.618 inches

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