Sebaran kondisional

Diberekeun dua sebaran gabungan variabel random X jeung Y, sebaran kondisional probabiliti 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 pY|X(y | x) and this is

where pX,Y(x, y) gives the joint distribution of X and Y, while pX(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 pY|X(y | x) = pY(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, so that the sum over all x need not be 1.

Sebaran probabilitas