Joint Probability Density Function
A joint probability density function for two random variables X and Y is defined by:
f (x, y) = Pr[(X = x) and (Y = y)]
- f(x, y) = 0 for values of x and y, which cannot serve as possible results for X and Y
- Sum of all possible values of f(x, y) must equal 1 (Since sum of probabilities for all possible events must equaly unity)
Example:
Determine the joint probability densitiy function for discrete random variables variables X and Y representing the top and bottom numbers of a fair die when tossed.
Solution:
Since the die being tossed is fair and sum of top and bottom numbers of a die always equal seven, a table containing respective probabilies between the two discrete random variables X and Y may be constructed:
| Y = 1 | Y = 2 | Y = 3 | Y = 4 | Y = 5 | Y = 6 | |
| X = 1 | 0 | 0 | 0 | 0 | 0 | 1/6 |
| X = 2 | 0 | 0 | 0 | 0 | 1/6 | 0 |
| X = 3 | 0 | 0 | 0 | 1/6 | 0 | 0 |
| X = 4 | 0 | 0 | 1/6 | 0 | 0 | 0 |
| X = 5 | 0 | 1/6 | 0 | 0 | 0 | 0 |
| X = 6 | 1/6 | 0 | 0 | 0 | 0 | 0 |
Using the above table, it follows that:
