In probability theory and statistics, the conditional probability distribution is a probability distribution that describes the probability of an outcome given the occurrence of a particular event. Given two jointly distributed random variables, the conditional probability distribution is the probability distribution are categorical variables, a conditional probability table is typically used to represent the conditional probability. The conditional distribution contrasts with the marginal distribution of a random variable, which is its distribution without reference to the value of the other variable.

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 referred to by corresponding names such as the conditional mean and conditional variance.

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.

Conditional discrete distributions

For discrete random variables, the conditional probability mass function can be written according to its definition as:

is:

Example

otherwise. is 1/3 (since there are three possible prime number rolls—2, 3, and 5—of which one is even).

Conditional continuous distributions

Similarly for continuous random variables, the conditional probability density function can be written as

gives the joint density gives the marginal density.

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.

Example

The graph shows a bivariate normal joint density plane.

Relation to independence

are independent if and only if, having a joint density function.

Properties

, it is a likelihood function need not be 1.

.

Measure-theoretic formulation

be a probability space, the Radon–Nikodym theorem implies that there is, called the conditional probability, and such a random variable is uniquely defined up to sets of probability zero. A conditional probability is called regular is a probability measure a.e.

Special cases:

• , the indicator function (defined below).

is called the conditional probability distribution, then it is called regular.

), every conditional probability distribution is regular. almost surely.

Relation to conditional expectation

, 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:

is a version of the conditional expectation:

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

Interpretation of conditioning on a Sigma Field

.

occurring. This can be shown with a counter-example:

Consider a probability space on the unit interval.

See also

• Conditioning (probability)

• Conditional probability

• Regular conditional probability

• Bayes' theorem

References

Citations

Sources