Conditional probability distribution: Difference between revisions

Revision as of 12:58, 24 March 2010

Revision as of 02:37, 30 March 2010

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Given two jointly distributed [[random variable]]s ''X'' and ''Y'', the '''conditional probability distribution''' of ''Y'' given ''X'' is the [[probability distribution]] of ''Y'' when ''X'' is known to be a particular value.

For [[discrete random variable]]s, the [[conditional probability]] mass function of ''Y'' given (the occurenceoccurrence of) the value ''x'' of ''X'', can be written, using the definition of [[conditional probability]], as:

:<math>p_Y(y\mid X = x)=P(Y = y \mid X = x) = \frac{P(X=x\ \cap Y=y)}{P(X=x)}.</math>

As seen from the definition, and due to its occurenceoccurrence, it is necessary that <math>P(X=x) > 0.</math>

The relation with the probability distribution of ''X'' given ''Y'' is:

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:<math>P(Y=y \mid X=x) P(X=x) = P(X=x\ \cap Y=y) = P(X=x \mid Y=y)P(Y=y).</math>

Similarly for [[continuous random variable]]s, the conditional [[probability density function]] of ''Y'' given (the occurenceoccurrence of) the value ''x'' of ''X'', can be written as

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

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The relation with the probability distribution of ''X'' given ''Y'' is given by:

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

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.