Page 63 - DMTH404_STATISTICS

Unit 5: Functions of Random Variables

Let us start with a random vector (X,Y).B y definition X and Y are random variables defined on Notes
the sample space S of some experiment and each of which assigns a real number to every s  S.
Let g(x,y) be a real-valued function defined on R x R. Then the composite function Z = g(X,Y)
defined by
Z(s) = g [X(s), Y(s)], s  S

assignes to every outcome s  S a real number. Z is called a function of the random vector (X,Y).
For example, if g(x, y) = x + y, then we get Z = X + Y and if g(x,y) =xy, then we get Z = XY and so
on.

Now let us see how do we find the distribution function of Z. As in the univariate case, we shall
restrict ourselves to the continuous case. Here we shall discuss two methods for obtaining
distribution functions - Direct Method and Transformation Method. We shall first discuss Direct
Method.

5.1.1 Direct Method

Let (X,Y) be a random vector with the joint density function fx, y (x,y). Let g(x,y) be a real-valued
function defined on R x R. For z E 2, define
D = {(x, Y) : g(x, y)  z}
z
Then the distribution function of Z is defined as

P [Z z] =   f X,Y (X,Y) dx dy ...(1)

D z
Theoretically it is not difficult to write down the distribution function using (1). But in actual
practise it is sometimes difficult to evaluate the double integral.
We shall now illustrate the computation of distribution functions in the following examples.

Example 1: Suppose (X,Y) has the uniform distribution on [0,1] × [0,1] the unit square.
Then the joint density of (X,Y) is

 1 if 0  x, y  1
fX,y(x,y)  
 0 otherwise
Let us find the distribution function of Z = g(X,Y) = XY.

From the definition of a distrihution function of Z, we have
Fz (z) = P [XY Z]

z
=   fX,Y(x,y)dxdy if 0   1
D z

LOVELY PROFESSIONAL UNIVERSITY 55