# Bivariate Distributions: Definition & Examples

## What Is a Bivariate Distribution

A **bivariate distribution**, put simply, is the probability that a certain event will occur when there are two independent random variables in your scenario. For example, having two bowls, each filled with two different types of candies, and pulling one candy from each bowl gives you two independent random variables: the two different candies.

Since you're pulling one candy from each bowl at the same time, you have a bivariate distribution when calculating your probability of ending up with particular kinds of candies.

## What Does it Look Like?

So what does a bivariate distribution look like? Such a distribution actually doesn't have a standard look. You can create a table with these distributions or you can list each probability out one by one. In any case, there are always two independent random variables in any given scenario.

Here is what a bivariate distribution looks like in table form:

This bivariate distribution shows you the probability of picking red or blue candies from a red bowl and a blue bowl if you pick one candy from each bowl, and there are an equal number of red and blue candies in each bowl.

## Digging Deeper

Once you have your distribution, you can look at the numbers to see what your probabilities are for each possible option in your scenario. Looking at the table for the candies again, you see four options (two red candies, two candies from opposite colored bowls, two candies from same colored bowls, and two blue candies). The option for getting two red candies has a probability of one in four.

The last row (the one labeled Red Bowl) and the last column (the one labeled Blue Bowl) give you what is called the marginal probability distribution. So looking at the last row in the Red from Red Bowl column gives you the probability of you picking a red candy from the red bowl regardless of what happens when you pick from the blue bowl. As with all probability, everything has to add up to 1 as you can see by the 1 in the lower right.

## Example

Let's look at another probability. This bivariate distribution gives you the probabilities when you roll two fair dice.

According to this bivariate distribution, the probability to roll two ones with the two dice is 1 in 36. The probability to roll a three with dice A is one in six, regardless of what happens with dice B.

This example shows that a bivariate distribution table can have more than two options for a variable. Just because there are two variables doesn't mean each variable only has two choices.

## Lesson Summary

Let's review. **Bivariate distribution** are the probabilities that a certain event will occur when there are two independent random variables in your scenario. It can be in list form or table form, like this:

The distribution tells you the probability of each possible choice of your scenario. The last column and last row give you the marginal probability distribution of something happening to one variable regardless of what happens to the other variable.

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## Bivariate Distributions: True or False Activity

This activity will help assess your knowledge regarding the meaning and examples of a bivariate distribution.

Guidelines

Print or copy this page on a blank piece of paper. Based on the given scenario, identify whether the following statement is TRUE or FALSE. Neatly write your answers in the appropriate space provided.

*Blake tosses a pair of fair, eight-sided dice where one of the dice is BLUE, and the other is YELLOW. Below is a table displaying the probability of obtaining a pair of numbers after Blake rolls the two dice. Additionally, it also shows the probability of obtaining a number after rolling one dice.*

__________ 1. All probabilities listed in the table are correct.

__________ 2. The scenario above is an example of a bivariate distribution.

__________ 3. The sum of probabilities in rolling the blue dice should be equal to 2.

__________ 4. The sum of the probabilities in rolling the yellow dice should be equal to 1.

__________ 5. The probability of rolling (2,3) with the two dice is 1/48.

__________ 6. Rolling a 7 with the yellow dice has a chance of 1/8.

__________ 7. The two independent variables are the two numbers obtained from rolling two dice at the same time.

__________ 8. The probability of rolling the two dice to get a total of 8 is 8/64.

### Answer Key

1. FALSE, because the correct statement is: Some probabilities listed in the table are incorrect.

2. TRUE

3. FALSE, because the correct statement is: The sum of probabilities in rolling the blue dice should be equal to 1.

4. TRUE

5. FALSE, because the correct statement is: The probability of rolling (2,3) with the two dice is 1/64.

6. TRUE

7. TRUE

8. FALSE, because the correct statement is: The probability of rolling the two dice to get a total of 8 is 7/64.

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