< List of probability distributions < *Discrete Joint Distribution*

## What is a discrete joint distribution?

In a discrete joint distribution, the variable’s relationship can be dependent or independent:

**Independent**: one variable does not influence the other. For example, let’s say you flip a coin and roll a die. The probability of the coin landing on heads isn’t influenced by the probability of rolling a 6 on the die roll.**Dependent**: one variable influence the other. For example, you flip two coins and want to know the probability of both coming up heads.

## Discrete joint distribution example

With a joint probability mass function (PMF) we can calculate the probability of X and Y taking specific values at the same time.

The joint PMF of two discrete random variables *X* and *Y* is defined as

**P(X = x and Y = y),**

where *x* and* y* are certain values of the random variables X and Y.

For example, suppose we want to find the joint PMF for the sum of one red and one blue six-sided dice. Let’s assign X to the value of the red die’s random variable and Y to the blue die’s random variable.

Each die has six sides with values 1 through 6, so the possible values of X and Y are {1, 2, 3, 4, 5, 6}. To find the joint PMF, we need to calculate the probability of each possible sum of the red and blue dice.

One of the easiest ways to represent the joint PMF is with a table. In the following table, the rows correspond to the *X* values of and the columns correspond to the *Y* values.

X/Y |
1 |
2 |
3 |
4 |
5 |
6 |
---|---|---|---|---|---|---|

1 |
1/36 | 1/36 | 1/36 | 1/36 | 1/36 | 1/36 |

2 |
1/36 | 1/36 | 1/36 | 1/36 | 1/36 | 1/36 |

3 |
1/36 | 1/36 | 1/36 | 1/36 | 1/36 | 1/36 |

4 |
1/36 | 1/36 | 1/36 | 1/36 | 1/36 | 1/36 |

5 |
1/36 | 1/36 | 1/36 | 1/36 | 1/36 | 1/36 |

6 |
1/36 | 1/36 | 1/36 | 1/36 | 1/36 | 1/36 |

The table entries represent the probabilities of each sum. For example, if we wanted to know the probability of rolling a sum of 7 with the two dice, we would add up the probabilities of the outcomes (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1).

X/Y | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

1 |
1/36 | 1/36 | 1/36 | 1/36 | 1/36 | 1/36 |

2 |
1/36 | 1/36 | 1/36 | 1/36 | 1/36 |
1/36 |

3 |
1/36 | 1/36 | 1/36 | 1/36 |
1/36 | 1/36 |

4 |
1/36 | 1/36 | 1/36 |
1/36 | 1/36 | 1/36 |

5 |
1/36 | 1/36 |
1/36 | 1/36 | 1/36 | 1/36 |

6 |
1/36 |
1/36 | 1/36 | 1/36 | 1/36 | 1/36 |

Adding up the values, we get 6/36 = 1/6.

## Uses

Discrete joint distributions aren’t just a theoretical construct: they have real life uses in many fields such as data analysis and machine learning. As well as analyzing relationships, then can help us identify correlations and dependencies.

Specific real life examples:

- A discrete joint probability distribution can be used to find out if a patient is infected with a disease, given that they have a certain test result.
- It can be used to test the relationship between the number of goods sold in a certain day vs. the number of warranties.
- We could study the joint PMF of traffic at a remote location and traffic recorded by some imperfect traffic counter [1].

- MIT. Brief notes #4.