What is marginal probability?

Marginal probability is the probability of an event occurring in one variable, regardless of the value of the other variable.

In probability theory, a joint probability distribution is a probability distribution that describes the probability of two or more random variables. Marginal probability is the probability of an event occurring in one variable, regardless of the value of the other variable. It is obtained by summing or integrating the joint probability distribution over the values of the other variable.

For example, suppose we have two random variables X and Y with a joint probability distribution given by the following table:

| X/Y | 0 | 1 | 2 |
| --- | --- | --- | --- |
| 0 | 0.1 | 0.2 | 0.1 |
| 1 | 0.2 | 0.3 | 0.1 |
| 2 | 0.1 | 0.1 | 0.1 |

The marginal probability distribution of X is obtained by summing the joint probability distribution over the values of Y:

P(X=0) = 0.1 + 0.2 + 0.1 = 0.4
P(X=1) = 0.2 + 0.3 + 0.1 = 0.6
P(X=2) = 0.1 + 0.1 + 0.1 = 0.3

Similarly, the marginal probability distribution of Y is obtained by summing the joint probability distribution over the values of X:

P(Y=0) = 0.1 + 0.2 + 0.1 = 0.4
P(Y=1) = 0.2 + 0.3 + 0.1 = 0.6
P(Y=2) = 0.1 + 0.1 + 0.1 = 0.3

Marginal probability is useful in many areas of probability theory and statistics, such as hypothesis testing, Bayesian inference, and machine learning.