# Conditional probability distribution

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Given two jointly distributed random variables *X* and *Y*, the **conditional probability distribution** of *Y* given *X* is the probability distribution of *Y* when *X* is known to be a particular value. If the conditional distribution of *Y* given *X* is a continuous distribution, then its probability density function is known as the **conditional density function**.

The properties of a conditional distribution, such as the moments, are often called by corresponding names such as the conditional mean and conditional variance.

## Discrete distributions

For discrete random variables, the conditional probability mass function of *Y* given (the occurrence of) the value *x* of *X*, can be written, using the definition of conditional probability, as:

As seen from the definition, and due to its occurrence, it is necessary that

The relation with the probability distribution of *X* given *Y* is:

## Continuous distributions

Similarly for continuous random variables, the conditional probability density function of *Y* given (the occurrence of) the value *x* of *X*, can be written as

where *f _{X,Y}*(

*x, y*) gives the joint density of

*X*and

*Y*, while

*f*(

_{X}*x*) gives the marginal density for

*X*. Also in this case it is necessary that .

The relation with the probability distribution of *X* given *Y* is given by:

The concept of the conditional distribution of a continuous random variable is not as intuitive as it might seem: Borel's paradox shows that conditional probability density functions need not be invariant under coordinate transformations.

## Relation to independence

Random variables *X*, *Y* are independent if and only if the conditional distribution of *Y* given *X* is equal to the unconditional distribution of *Y*. For discrete random variables: *P*(*Y* = *y* | *X* = *x*) = *P*(*Y* = *y*) for all relevant *x* and *y*. For continuous random variables having a joint density: *f*_{Y}(*y* | *X=x*) = *f*_{Y}(*y*) for all relevant x and y.

## Properties

Seen as a function of *y* for given *x*, *P*(*Y* = *y* | *X* = *x*) is a probability and so the sum over all *y* (or integral if it is a conditional probability density) is 1. Seen as a function of *x* for given *y*, it is a likelihood function, so that the sum over all *x* need not be 1.

## See also

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