Conditional probability distribution

Given two jointly distributed random variables

is known to be a particular value; in some cases the conditional probabilities may be expressed as functions containing the unspecified value

are categorical variables, a conditional probability table is typically used to represent the conditional probability.

is a continuous distribution, then its probability density function is known as the conditional density function.

More generally, one can refer to the conditional distribution of a subset of a set of more than two variables; this conditional distribution is contingent on the values of all the remaining variables, and if more than one variable is included in the subset then this conditional distribution is the conditional joint distribution of the included variables.

For discrete random variables, the conditional probability mass function of

can be written according to its definition as: Due to the occurrence of

in the denominator, this is defined only for non-zero (hence strictly positive)

Similarly for continuous random variables, the conditional probability density function of

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.

The graph shows a bivariate normal joint density for random variables

The intersection of that plane with the joint normal density, once rescaled to give unit area under the intersection, is the relevant conditional density of

For discrete random variables this means

, having a joint density function, it means

is a probability mass function and so the sum over all

(or integral if it is a conditional probability density) is 1.

, it is a likelihood function, so that the sum (or integral) over all

Additionally, a marginal of a joint distribution can be expressed as the expectation of the corresponding conditional distribution.

, and such a random variable is uniquely defined up to sets of probability zero.

A conditional probability is called regular if

is called the conditional probability distribution of

For a real-valued random variable (with respect to the Borel

), every conditional probability distribution is regular.

, define the indicator function: which is a random variable.

Note that the expectation of this random variable is equal to the probability of A itself: Given a

is a version of the conditional expectation of the indicator function for

: An expectation of a random variable with respect to a regular conditional probability is equal to its conditional expectation.

can be loosely interpreted as containing a subset of the information in

It is incorrect to conclude in general that the information in

This can be shown with a counter-example: Consider a probability space on the unit interval,