Conditional probability distribution

Given two jointly distributed random variables 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 variables, the conditional probability mass function of Y given (the occurrence of) the value x of X, can be written, using the definition of conditional probability, as:

${\displaystyle p_{Y}(y\mid X=x)=P(Y=y\mid X=x)={\frac {P(X=x\ \cap Y=y)}{P(X=x)}}.}$

As seen from the definition, and due to its occurrence, it is necessary that ${\displaystyle P(X=x)>0.}$

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

${\displaystyle P(Y=y\mid X=x)P(X=x)=P(X=x\ \cap Y=y)=P(X=x\mid Y=y)P(Y=y).}$

Similarly for continuous random variables, the conditional probability density function of Y given (the occurrence of) the value x of X, can be written as

${\displaystyle f_{Y}(y\mid X=x)={\frac {f_{X,Y}(x,y)}{f_{X}(x)}},}$

where f_X,Y(x, y) gives the joint density of X and Y, while f_X(x) gives the marginal density for X. Also in this case it is necessary that ${\displaystyle f_{X}(x)>0}$.

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

${\displaystyle f_{Y}(y\mid X=x)f_{X}(x)=f_{X,Y}(x,y)=f_{X}(x\mid Y=y)f_{Y}(y).}$

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(y | X=x) = f_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 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.

References

• Taboga, Marco (2010). "Fundamentals of probability. Conditional probability distributions" (Document). {{cite document}}: Cite document requires |publisher= (help); Unknown parameter |url= ignored (help)

See also

• Conditioning (probability)
• Conditional probability
• Conditional expectation
• Regular conditional probability