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714
Probability and Random Variables
Appendix B
Example B–11 PDF FOR THE SUM OF TWO RANDOM VARIABLES
Suppose that we have a circuit configuration (such as an operational amplifier) that sums two
inputs—x 1 and x 2 —to produce the output
y = A(x 1 + x 2 )
(B–102)
where A is the gain of the circuit. Assume that f(x 1 , x 2 ) is known and that we wish to obtain a
formula for the PDF of the output in terms of the joint PDF for the inputs.
We can use the theorem described by Eq. (B–99) to solve this problem. However, for two
inputs, we need two outputs in order to satisfy the assumptions of the theorem. This is achieved
by defining an auxiliary variable for the output, (say, y 2 ). Thus,
y 1 = h 1 (x) = A(x 1 + x 2 )
(B–103)
y 2 = h 2 (x) = Ax 1
(B–104)
The choice of the equation to use for the auxiliary variable, Eq. (B–104), is immaterial, provided
that it is an independent equation so that the determinant, J(yx), is not zero. However, the equation
is usually selected to simplify the ensuing mathematics. Using Eqs. (B–103) and (B–104), we get
J = Det c A A
A 0 d = -A2
Substituting this into Eq. (B–99) yields
or
(B–105)
f y (y 1 , y 2 ) = 1 (B–106)
A 2 f xa y 2
A , 1
A (y 1 - y 2 )b
We want to find a formula for f y1 (y 1 ), since y 1 = A(x 1 + x 2 ). This is obtained by evaluating
the marginal PDF from Eq. (B–106).
or
f y (y 1 , y 2 ) = f x(x 1 , x 2 )
|-A 2 |
q
f y1 (y 1 ) = f y (y 1 , y 2 ) dy 2
L
-q
q
f y1 (y 1 ) = 1 (B–107)
A 2 f x a y 2
L-q
A , 1
A (y 1 - y 2 )bdy 2
This general result relates the PDF of y = y 1 to the joint PDF of x where y = A(x 1 + x 2 ). If x 1 and
x 2 are independent and A = 1, Eq. (B–107) becomes
q
f(y) = f x1 (l)f x2 (y - l) dl
L
-q
`
x = h -1 (y)
or
f(y) = f x1 (y) * f x2 (y)
(B–108)
where * denotes the convolution operation. Similarly, if we sum N independent random variables,
the PDF for the sum is the (N - 1)-fold convolution of the one-dimensional PDFs for the
N random variables.