Page 185 - Fundamentals of Probability and Statistics for Engineers

168 Fundamentals of Probability and Statistics for Engineers

p (k)
X

p

qp
2
q p
k
1 2 3 4 5 6 7 ...

Figure 6.1 Geometric distribution p (k)
X
The corresponding probability distribution function is

m x
X m 1
F X x†ˆ p k†ˆ p ‡ qp ‡ ‡ q p
X
kˆ1
2
m
ˆ 1 q† 1 ‡ q ‡ q ‡ ‡ q m 1 †ˆ 1 q ; 6:15†
where m is the largest integer less than or equal to x. The mean and variance of
X can be found as follows:
1 1
X k 1 X d k
EfXgˆ kq p ˆ p q
dq
kˆ1 kˆ1
1
d X k d q 1
ˆ p q ˆ p ˆ : 6:16†
dq dq 1 q p
kˆ1
In the above, the interchange of summation and differentiation is allowed
because jqj < 1. Following the same procedure, the variance has the form

1 p
2
ˆ : 6:17†
X 2
p
Example 6.5. Problem: a driver is eagerly eyeing a precious parking space
some distance down the street. There are five cars in front of the driver, each of
which having a probability 0.2 of taking the space. What is the probability that
the car immediately ahead will enter the parking space?
Answer: for this problem, we have a geometric distribution and need to
evaluate p (k) for k ˆ 5 and p ˆ 0:2. Thus,
X
4
p 5†ˆ 0:8† 0:2†ˆ 0:82;
X

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