Page 176 - Fundamentals of Probability and Statistics for Engineers

Functions of Random Variables 159

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 ˆ 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 ˆjX 1
X 2 j where X 1 and X 2 are independent random
variables with respective pdfs f (x 1 ) and f (x 2 ).
X 1 X 2
5.24 The light intensity I at a given point X distance away from a light source is
I ˆ C/X where C is the source candlepower. Determine the pdf of I if the pdfs
2
of C and X are given by
1
8
< ; for 64 c 100;
f c†ˆ 36
C
:
0; elsewhere;

1; for 1 x 2;
f x†ˆ
X
0; elsewhere;
and C and X are independent.
5.25 Let X 1 and X 2 be independent and identically distributed according to
1
x 2
f x 1 †ˆ exp 1 ;
1 < x 1 < 1;
X 1 1=2
2 † 2
and similarly for X 2 . By means of techniques developed in Section 5.2, determine
2 1/2
the pdf of Y , where Y ˆ (X ‡ X ) . Check your answer with the result obtained
2
1 2
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
2
2 1/2
2
distributed random variables, that is, Y ˆ (X ‡ X ‡ X ) . (Hint: use spherical
1 2 3
coordinates to carry out integration.)
5.27 The joint probability density function (jpdf) of random variables X 1 , X 2 , and X 3
takes the form
6
8
>
< 4 ; for x 1 ; x 2 ; x 3 † > 0; 0; 0†;
f x 1 ; x 2 ; x 3 †ˆ 1 ‡ x 1 ‡ x 2 ‡ x 3 †
X 1 X 2 X 3
>
0; elsewhere:
:
Find the pdf of Y ˆ X 1 ‡ X 2 ‡ X 3 .
5.28 The pdfs of two independent random variables X 1 and X 2 are

e
x 1 ; for x 1 > 0;
f x 1 †ˆ
X 1
0; for x 1 0;

e
x 2 ; for x 2 > 0;
f x 2 †ˆ
X 2
0; for x 2 0:

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