## 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: