CONDITIONAL PROBABILITY DISTRIBUTION

Given two jointly distributed random variables ''X'' and ''Y'', the 'conditional probability distribution' of ''Y'' given ''X'' (written "''Y'' | ''X''") is the probability distribution of ''Y'' when ''X'' is known to be a particular value.
For discrete random variables, the conditional probability mass function can be written as ''P''(''Y'' = ''y'' | ''X'' = ''x''). From the definition of conditional probability, this is
:
Similarly for continuous random variables, the conditional probability density function can be written as ''p''_''Y''|''X''(''y'' | ''x'') and this is
:
where ''p''_''X'',''Y''(x, y) gives the joint distribution of ''X'' and ''Y'', while ''p''_''X''(''x'') gives the marginal distribution for ''X''.
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.
If for discrete random variables ''P''(''Y'' = ''y'' | ''X'' = ''x'') = ''P''(''Y'' = ''y'') for all ''x'' and ''y'', or for continuous random variables ''p''_''Y''|''X''(''y'' | ''x'') = ''p''_''Y''(''y'') for all x and y, then ''Y'' is said to be independent of ''X'' (and this implies that ''X'' is also independent of ''Y'').
Seen as a function of ''y'' for given ''x'', ''P''(''Y'' = ''y'' | ''X'' = ''x'') is a probability and so the sum over all ''y'' (or integral if it is a density) is 1. Seen as a function of ''x'' for given ''y'', it is a likelihood function, so that the sum over all ''x'' need not be 1.

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See also

See also

★ Conditional probability

★ Conditional expectation