Conditional probability distribution

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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 ${\displaystyle X}$ and ${\displaystyle Y}$, the conditional probability distribution of ${\displaystyle Y}$ given ${\displaystyle X}$ is the probability distribution of ${\displaystyle Y}$ when ${\displaystyle X}$ is known to be a particular value; in some cases the conditional probabilities may be expressed as functions containing the unspecified value ${\displaystyle x}$ of ${\displaystyle X}$ as a parameter. When both ${\displaystyle X}$ and ${\displaystyle Y}$ 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.

If the conditional distribution of ${\displaystyle Y}$ given ${\displaystyle X}$ is a continuous distribution, then its probability density function is known as the conditional density function.^[1] 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.