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B–8 Functional Transformations of Random Variables 705
y= h(x) y= A sin x
A
y 0
y
A
y 0
∏
∏
2
∏
2
∏
x
x 1 x 2
A A
f y (y)
–A
A
f x (x)
1
2∏
∏
x 1 x 2 ∏
x
y 0 y 0
x 2 x 1
A 2 2
y 0
A 2 2
y 0
Figure B–9
Evaluation of the PDF of a sinusoid (Ex. B–5).
namely, x 1 and x 2 , as shown in the figure. Thus, M = 2, provided that ƒyƒ 6 A. Otherwise, M = 0.
Evaluating the derivative of Eq. (B–69), we obtain
and for 0 … y … A, we get
and
dy
dx = A cos x
x 1 = Sin -1 a y A b
x 2 = p - x 1
where the uppercase S in Sin -1 (·) denotes the principal angle. A similar result is obtained for
-A … y … 0. Using Eq. (B–68), we find that the PDF for y is
f x (x 1 ) f x (x 2 )
+
, ƒ y ƒ … A
ƒ A cos x
f y (y) = µ 1 ƒ ƒ -A cos x 2 ƒ
0, y elsewhere
(B–70)
The denominators of these two terms are evaluated with the aid of the triangles shown in the insert
of Fig. B–9. By using this result and substituting the uniform PDF for f x (x), (B–70) becomes