Page 130 - Schaum's Outlines - Probability, Random Variables And Random Processes
P. 130
CHAP. 41 FUNCTIONS OF RANDOM VARIABLES, EXPECTATION, LIMIT THEOREMS 123
then
where g'(x) is the derivative of g(x).
4.3. FUNCTIONS OF TWO RANDOM VARIABLES
A. One Function of Two Random Variables:
Given two r.v.'s X and Y and a function g(x, y), the expression
defines a new r.v. 2. With z a given number, we denote Dz the subset of Rxy [range of (X, Y)] such
that g(x, y) 5 z. Then
where ((X, Y) E DZ) is the event consisting of all outcomes such that the point {X([), Y(5)) E DZ.
Hence
FAz) = P(Z I z) = P[g(X, Y) 5 z] = P((X, Y) E D,} (4.1 1)
If X and Y are continuous r.v.'s with joint pdffxAx, y), then
B. Two Functions of Two Random Variables:
Given two r.v.'s X and Y and two functions g(x, y) and h(x, y), the expression
defines two new r.v.'s Z and W. With z and w two given numbers, we denote Dzw the subset of Rxy
[range of (X, Y)] such that g(x, y) < z and h(x, y) 5 w. Then
where ((X, Y) E Dzw} is the event consisting of all outcomes such that the point {X(c), Y([)} E DZw.
Hence
In the continuous case, we have
Determination of fz iw(z, w) jiom fdx, y) :
Let X and Y be two continuous r.v.'s with joint pdf fxy(x, y). If the transformation
z = g(x, Y) w = h(x, Y)
is one-to-one and has the inverse transformation
