Page 140 - Fundamentals of Probability and Statistics for Engineers

Functions of Random Variables 123

is the inverse function of g(x), or the solution for x in Equation (5.5) in terms of
y. Hence,

1

1
F Y y†ˆ P Y y†ˆ P‰g X† yŠˆ P‰X g y†Š ˆ F X ‰g y†Š: 5:7†
Equation (5.7) gives the relationship between the PDF of X and that of Y , our
desired result.
The relationship between the pdfs of X and Y are obtained by differentiating
both sides of Equation (5.7) with respect to y. We have:

1
dF Y y† d
1
1 dg y†
f y†ˆ ˆ fF X ‰g y†Šg ˆ f ‰g y†Š : 5:8†
Y X
dy dy dy
It is clear that Equations (5.7) and (5.8) hold not only for the particular
transformation given by Equation (5.5) but for all continuous g(x) that are strictly
monotonic increasing functions of x, that is, g(x 2 )> g(x 1 ) whenever x 2 > x 1 .
Consider now a slightly different situation in which the transformation is
given by
Y ˆ g X†ˆ
2X ‡ 1: 5:9†

Starting again with F Y (y) ˆ P(Y y), and reasoning as before, the region

1
Y y in the range space R Y is now mapped into the region X > g (y), as
indicated in Figure 5.3. Hence, we have in this case

1
F Y y†ˆ P Y y†ˆ P‰X > g y†Š
5:10†

1

1
ˆ 1
P‰X g y†Š ˆ 1
F X ‰g y†Š:
y

y =–2x +1

y

x

1– y
–1
x = g (y)=
2
Figure 5.3 Transformation defined by Equation (5.9)

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