Fundamentals of Probability and Statistics for Engineers

For all y, Equation (5.25) yields

a result identical to the solution for Example 5.4 [see Equation (5.18)].

Ex ample 5. 7. Problem: determine the pdf of Y X^2 where X is normally
distributed according to

As shown in Figure 5.13, fY (y) 0 for y < 0 since the transformation
equation has no real roots in this range. For y 0, the two roots of
2

y

y = x^2

y

x

Figure 5. 13 Transformation y x^2

132 Fundamentals of Probability and Statistics for Engineers

ˆ



x 1 ; 2 ˆg^1 ;^12 …y†ˆy^1 =^2 :

x 2 =–√y x 1 =√y

ˆ

fY…y†ˆ

X^2

jˆ 1

f‰g^ j^1 …y†Š

dg^ j^1 …y†

dy

ˆ

1

2 

1

1 ‡y^2



‡

1

2 

1

1 ‡y^2



ˆ

1

… 1 ‡y^2 †

; 

1 <y< 1 ;

… 5 : 26 †

ˆ

fX…x†ˆ

1

… 2 †^1 =^2

e^ x

(^2) = 2
;
1 <x< 1 : … 5 : 27 †
yˆxare