Fundamentals of Probability and Statistics for Engineers

.It exists for discrete and continuous random variables and has values between

0and1.
.It is a nonnegative, continuous-to-the-left, and nondecreasing function of the

real variable Moreover, we have

.If and are two real numbers such that , then

This relation is a direct result of the identity

We see from Equation (3.3) that the probability of having a value in an

arbitrary interval can be represented by the difference between two values of

the PDF. Generalizing, probabilities associated with any sets of intervals are

derivable from the PDF.

Example 3.1.Let a discrete random variable assume values 1, 1, 2, and 3,

with probabilities^14 ,^18 ,^18 ,and^12 , respectively. We then have

This function is plotted in Figure 3.1. It is typical of PDFs associated with

discrete random variables, increasing from 0 to 1 in a ‘staircase’ fashion.

A continuous random variable assumes a nonenumerable number of values

over the real line. Hence, the probability of a continuous random variable

assuming any particular value is zero and therefore no discrete jumps are

possible for its PDF. A typical PDF for continuous random variables is

shown in Figure 3.2. It has no jumps or discontinuities as in the case of the

discrete random variable. The probability of having a value in a given

interval is found by using Equation (3.3), and it makes sense to speak only of

this kind of probability for continuous random variables. For example, in

F igure 3.2.

Clearly, for any.

40 Fundamentals of Probability and Statistics for Engineers

x.

FX…1†ˆ 0 ; and FX…‡1†ˆ 1 : … 3 : 2 †

a b a<b

P…a<Xb†ˆFX…b†FX…a†: … 3 : 3 †

P…Xb†ˆP…Xa†‡P…a<Xb†:

X

X 

FX…x†ˆ

0 ; for x< 1 ;

1

4

; for  1 x< 1 ;

3

8 ; for 1x<^2 ;

1

2

; for 2x< 3 ;

1 ; for x 3 :

8

>>

>>

>>

>>

<

>>>

>>

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>:

X

P… 1 <X 1 †ˆFX… 1 †FX… 1 †ˆ 0 : 8  0 : 4 ˆ 0 : 4 :

P 9 Xˆa)ˆ 0 a