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S I N M A P
U S E R ’ S
F(a ) =
M A N U A L
–
A P P E N D I X
1
a ln( b 2 y 2 / a ) + a + b1 y1 − b 1 y 2 − b 2 y1
( y 2 − y1 )(b 2 − b1 )
if a > y2b2
F(a) = 1
This defines a generic function, which with parameters is denoted
FA(y1, y2, b1, b2; a) = Prob (A < a)
(A-27)
a = yb
y2
y1
b1
b2
FIGURE A-2. Evaluation of CDF for product of uniform random variables
Now the combination given by (A-25) has density function the convolution of the
density functions of A and X, namely
f Z ( z) =
∞
∫f
A
(a ) f X (z − a )da
(A-28)
−∞
Since X ~ U(x1, x2) this can be written
f Z ( z) =
z − x1
∫
z−x2
F (z − x 1 ) − FA (z − x 2 )
f A (a )
da = A
x 2 − x1
x 2 − x1
(A-29)
To get the cumulative distribution function of Z we need the integral of FA(a), which
will be, denoted FAI(a). Then
FZ (z) =
FAI (z − x 1 ) − FAI (z − x 2 )
x 2 − x1
(A-30)
Integrating FA(a), the expressions between equations (A-26) and (A-27) we obtain
if a < y1b1 = a1
FAI(a) = C1 = FAI1(a) + C1
if y1b1 = a2 < a < y1b2 = a3
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