Fundamentals of Probability and Statistics for Engineers

For part (b),

Figure 3.9 shows pX (x) for the discrete portion and fX (x) for the continuous
portion of X. They are given by:

and

;

Note again that the area under fX (x) is no longer one but is

To obtain P(X 2) and P(2 X 6), both the discrete and continuous
portions come into play, and we have, for part (a),

3 x

pX(x)

fX(x)

3 x

—^1

2e

—^1

3

—^1

3e

—^1

6e

(a) (b)

Figure 3. 9 (a) Partial probability mass function, pX (x), and (b) partial probability

density function, fX (x), of X, as described in Example 3.4

48 Fundamentals of Probability and Statistics for Engineers

P… 2 <X 6 †ˆFX… 6 †FX… 2 †

ˆ 1 

e^2

2



… 1 e^2 =^3 †ˆe^2 =^3 

e^2

2

:

pX…x†ˆ

1

2e

; atxˆ 3 ;

0 ; elsewhere;

8

<

:

fX…x†ˆ

dFX…x†

dx

ˆ

0 ; forx< 0 ;

1

3

ex=^3 ; for 0x<

1

6

ex=^3 ; forx 3 :

8

>>

>>

<

>>>

>:

3

1 pX… 3 †ˆ 1 

1

2e

:

> < 

P…X> 2 †ˆ

Z 1

2

fX…x†dx‡pX… 3 †

ˆ

1

3

Z 3

2

ex=^3 dx‡

1

6

Z 1

3

ex=^3 dx‡

1

2e

ˆe^2 =^3