Conditional probability distribution: Difference between revisions

Revision as of 12:59, 22 January 2010

Revision as of 22:21, 22 January 2010

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Similarly for [[continuous random variable]]s, the conditional [[probability density function]] can be written as ''f''_''Y''(''y'' | ''X=x'') and this is

:<math>f_{Yf_Y(y \mid X=x}(y) = \frac{f_{X, Y}(x, y)}{f_X(x)}= \frac{f_{Xf_X(x \mid Y=y}(x)f_Y(y)}{f_X(x)}, </math>

where ''f''_''X'',''Y''(x, y) gives the [[joint distribution|joint density]] of ''X'' and ''Y'', while ''f''_''X''(''x'') gives the [[marginal density]] for ''X''.

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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 ''f''_{''Y'' | ''X=x''}(''y'' | ''X=x'') = ''f''_''Y''(''y'') for all x and y, then ''Y'' is said to be [[Statistical independence|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 conditional probability 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.