Fundamentals of Probability and Statistics for Engineers

a case in which random variable X is continuously distributed over the real line
except at X 0, where P(X 0) is a positi ve quantity. This situation may arise
when, for example, random variable X represents the waiting time of a customer
at a ticket counter. Let X be the time interval from time of arrival at the ticket
counter to the time being served. It is reasonable to expect that X will assume
values over the interval X 0. At X 0, however, there is a finite probability of
no t ha vin g t o wait a t a ll, giving r ise t o th e sit ua tio n d epict ed in F igur e 3.7.
Strictly speaking, neither a pmf nor a pdf exists for a random variable of the
mixed type. We can, however, still use them separately for different portions of
the distribution, for computational purposes. Let fX (x) be the pdf for the
continuous portion of the distribution. It can be used for calculating probabil-
ities in the positive range of x values for this example. We observe that the total
area under the pdf curve is no longer 1 but is equal to 1 P(X 0).

Example 3.4.Problem: since it is more economical to limit long-distance
telephone calls to three minutes or less, the PDF of X – the duration in minutes
of long-distance calls – may be of the form

Determine the probability that X is (a) more than two minutes and (b) be tween
two and six minutes.
Answer: the PDF of X is plotted in Figure 3.8, showing that X has a mixed-
type distribution. The desired probabilities can be found from the PDF as
before. Hence, for part (a),

x

3

FX(x)

1– e–½

1– e–1

1

Figure 3. 8 Probability distribution function, FX (x), of X, as described in Example 3.4

Random Variables and Probability D istributions 47

ˆ ˆ

 ˆ

 ˆ

FX…x†ˆ

0 ; forx< 0 ;

1 ex=^3 ; for 0x< 3 ;

1 e

x= 3

2 ; forx^3 :

8

<

:

P…X> 2 †ˆ 1 P…X 2 †ˆ 1 FX… 2 †

ˆ 1 … 1 e^2 =^3 †ˆe^2 =^3 :