A **bivariate distribution** (or *bivariate probability distribution*) is a joint distribution with two variables of interest. The bivariate distribution gives probabilities for simultaneous outcomes of the two random variables.

- A
**discrete joint distribution**has at most a countable number of possible outcomes (basically, “countable” means you can label the numbers (like if you’re counting whole numbers 1, 2, 3…). - A
**continuous joint distribution**can be described by a non-negative function [1].

If the two variables are obtained through random sampling, you may be able to find correlations or run a regression analysis to investigate the link between the two variables or to predict future (or past) behavior of those variables.

Bivariate distributions are quite common in real life. For example:

- In an annual checkup,
**cholesterol levels and triglyceride levels**might be combined to measure heart health. - A gambler might want to know
**the probability of rolling a six, and then another six.** - A cafÃ© might be interested in studying
**coffee sales vs. coffee cake sales**. - In transportation,
**the role of alcohol in car crashes**is often investigated.

## Table Representation of Bivariate Distribution

Unlike many probability distributions, a bivariate distribution doesn’t have a unified look; each distribution is different and it depends on what variables you are studying.

A table of rows and columns can be used to display discrete random variables with a finite (fixed) number of values. The rows represent one of the variables, the columns the other. For example, the following table shows the bivariate distribution of tossing a tails if you flip a coin three times:

- The rows represent the X-values, which in this examples is the total number of tosses.
- The columns represent the Y-variable, which is the number of tosses to get the first tails (if no tails occur, this is set to zero).
- Each intersection represents an X-Y combination; the probabilities in those cells represent the joint probabilities.

As this is a probability distribution, the probabilities in all the cells add up to 1 [2].

## References

[1] Liu, M. STA 611: Introduction to Mathematical Statistics; Lecture 4: Random Variables and Distributions. Retrieved November 10, 2021 from: http://www2.stat.duke.edu/courses/Fall18/sta611.01/Lecture/Lecture04.pdf

[2] Gehrman, J. 6. Bivariate Rand. Vars. Bivariate Probability Distributions. Retrieved November 10, 2021 from: https://www.csus.edu/indiv/j/jgehrman/courses/stat50/bivariate/6bivarrvs.htm