Fundamentals of Probability and Statistics for Engineers

Let us compute some of the probabilities using The probability
(0 1) is numerically equal to the area under from 0 to
1, as shown in F igure 3.6(a). It is given by

The probability is obtained by computing the areaunder to the
right of 3. H ence,

The same probabilities can be obtained from by taking appropriate
differences, giving:

Let us note that there is no numerical difference between (0 1) and

(0 1) for continuous random variables, since ( 0) 0.

3.2.4 M ixed-Type D istribution

There are situations in which one encounters a random variable that is partially
discrete and partially continuous. The PDF given in Figure 3.7 represents such

1

0

FX(x)

x

Figure 3.7 A mixed-type probability distribution function,

46 Fundamentals of Probability and Statistics for Engineers

FX…x†ˆ

Z x

1

fX…u†duˆ 0 ; forx< 0 ;

1 eax; forx0.

8

<

:

… 3 : 15 †

fX 9 x).
P <X fX 9 x) xˆ
xˆ

P… 0 <X 1 †ˆ

Z 1

0

fX…x†dxˆ 1 ea:

P 9 X>3) fX 9 x)

xˆ

P…X> 3 †ˆ

Z 1

3

fX…x†dxˆe^3 a:

FX 9 x)

P… 0 <X 1 †ˆFX… 1 †FX… 0 †ˆ… 1 ea† 0 ˆ 1 ea;

P…X> 3 †ˆFX…1†FX… 3 †ˆ 1 … 1 e^3 a†ˆe^3 a:

P <X
P X P Xˆˆ

FX 9 x)