## Joint Probability Distribution, Probability

### Introduction

The joint probability distribution for X and Y defines the probability of events defined in terms of both X and Y. As defined in the form below:

where by the above represents the probability that event **x** and **y** occur at the same time.

The ** cumulative distribution function** for a joint probability distribution is given by:

In the case of only two random variables, this is called a bivariate distribution, but the concept generalises to any number of random variables, giving a multivariate distribution. The equation for joint probability is different for both dependent and independent events.

### Discrete Case

The joint probability function of two discrete random variables is equal to (Similar to Bayes' theorem):

In general, the joint probability distribution of n discrete random variables *X _{1} , X_{2} , ... ,X_{n}* is equal to:

This identity is known as the * chain rule of probability*.

Since these are probabilities, we have:

generalising for n discrete random variables *X _{1} , X_{2} , ... ,X_{n}* :

### Continuous Case

Similarly for continuous random variables, the joint probability density function can be written as f_{X,Y}(x, y) and this is :

where f_{Y|X}(y|x) and f_{X|Y}(x|y) give the conditional distributions of Y given X=x and of X given Y=y respectively, and f_{X}(x) and f_{Y}(y) give the marginal distributions for X and Y respectively.

Again, since these are probability distributions, one has:

### Mixed Case

In some situations X is continuous but Y is discrete. For example, in a logistic regression, one may wish to predict the probability of a binary outcome Y conditional on the value of a continuously distributed X. In this case, (X, Y) has neither a probability density function nor a probability mass function in the sense of the terms given above. On the other hand, a "mixed joint density" can be defined in either of two ways:

Formally, f_{X,Y}(x , y) is the probability density function of (X, Y) with respect to the product measure on the respective supports of X and Y. Either of these two decompositions can then be used to recover the joint cumulative distribution function:

The definition generalises to a mixture of arbitrary numbers of discrete and continuous random variables.

### References & Resources

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