Fundamentals of Probability and Statistics for Engineers

where is a constant. Suppose that demand X 2 at this location during the same

time interval has the same distribution as X 1 and is independent of X 1. Determine

the pdf of Y X 2 X 1 where Y represents the excess of taxis in this time interval

(positive and negative).

5.23 Determine the pdf of Y where X 1 and X 2 are independent random
variables with respective pdfs fX 1 (x 1 )andfX 2 (x 2 ).

5.24 The light intensity I at a given point X distance away from a light source is
2
of C and X are given by

and C and X are independent.

5.25 Let X 1 and X 2 be independent and identically distributed according to

and similarly for X 2. By means of techniques developed in Section 5.2, determine

the pdf of Y, where Y (X 12 X 22 )1/2. Check your answer with the result obtained

in Example 5.19. (Hint: use polar coordinates to carry out integration.)

5.26 Extend the result of Problem 5.25 to the case of three independent and identically
distributed random variables, that is, Y (X 12 X^22 X 32 )1/2. (Hint: use spherical
coordinates to carry out integration.)

5.27 The joint probability density function (j pdf) of ra ndom variables X 1 ,X 2 ,andX 3
takes the form

Find the pdf of Y X 1 X 2 X 3.

5.28 The pdfs of two independent random variables X 1 and X 2 are

Functions of Random Variables 159



ˆ

ˆjX 1

 X 2 j

ˆ

fC…c†ˆ

1

36

; for 64c 100 ;

0 ; elsewhere;

8

<

:

fX…x†ˆ

1 ; for 1x 2 ;

0 ; elsewhere;



fX 1 …x 1 †ˆ

1

… 2 †^1 =^2

exp

x^21

2



; 

1 <x 1 < 1 ;

ˆ‡

ˆ ‡‡

fX 1 X 2 X 3 …x 1 ;x 2 ;x 3 †ˆ

6

… 1 ‡x 1 ‡x 2 ‡x 3 †^4

;for…x 1 ;x 2 ;x 3 †>… 0 ; 0 ; 0 †;

0 ; elsewhere:

8

><

>:

‡‡

fX 1 …x 1 †ˆ

e^ x^1 ; forx 1 > 0 ;

0 ; forx 1 0;



fX 2 …x 2 †ˆ

e^ x^2 ; forx 2 > 0 ;

0 ; forx 2  0 :



IC/Xwhere C is the source candlepower. Determine the pdf of I if the pdfs

ˆ