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In [[probability theory]] and [[statistics]], given two [[joint probability distribution|jointly distributed]] [[random variable]]s <math>X</math> and <math>Y</math>, the '''conditional probability distribution''' of ''Y'' given ''X'' is the [[probability distribution]] of <math>Y</math> when <math>X</math> is known to be a particular value; in some cases the conditional probabilities may be expressed as functions containing the unspecified value <math>x</math> of <math>X</math> as a parameter. When both <math>X</math> and <math>Y</math> are [[categorical variable]]s, 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 <math>Y</math> given <math>X</math> is a [[continuous distribution]], then its [[probability density function]] is known as the '''conditional density function'''.<ref>{{cite book |first=Sheldon M. |last=Ross |authorlink=Sheldon M. Ross |title=Introduction to Probability Models |___location=San Diego |publisher=Academic Press |edition=Fifth |year=1993 |isbn=0-12-598455-3 |pages=88–91 }}</ref> The properties of a conditional distribution, such as the [[Moment (mathematics)|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.
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