3.9 Jointly Distributed Random Variables

Suppose we define two random variables X and Y on the sample space S. Thus each point in S has a value for X and a value for Y. Then X and Y are said to be jointly distributed.

If X and Y are both discrete random variables they have a joint probability function f ( x, y) = P( X = x, Y = y) with the properties:

1. f ( x, y) ? 0 for all x, y.

2. ? _{all x} ? _{all y} f ( x, y) = 1.

3. P( X ? x, Y ? y) = F( x, y) = ? _{s ? x} ? _{t ? y} f ( s, t).

If X and Y are both continuous random variables they have a joint probability density function f ( x, y) with the properties:

1. f ( x, y) ? 0 for all x, y.

2. 

3. 

It is often the case when X and Y are discrete that they only take each a small number of values. In that case, it is usually convenient to represent the joint distribution by a table.

Examples

1. Two discrete random variables.

   Let X be the...