Conditional probability distribution: Difference between revisions

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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 ''<math>Y''</math> given ''<math>X''</math> 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]].

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Due to the occurrence of <math>P(X=x)</math> in athe denominator, this is defined only for non-zero (hence strictly positive) <math>P(X=x).</math>

The relation with the probability distribution of <math>X</math> given <math>Y</math> is:

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===Example===

Consider the roll of a fair {{dice}} and let <math>X=1</math> if the number is even (i.e., 2, 4, or 6) and <math>X=0</math> otherwise. Furthermore, let <math>Y=1</math> if the number is prime (i.e., 2, 3, or 5) and <math>Y=0</math> otherwise.

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