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. A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ...

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 In mathematics, a random variable is discrete if its probability distribution is discrete; a discrete probability distribution is one that is fully characterized by a probability mass function. ... This article defines some terms which characterize probability distributions of two or more variables. ... This article defines some terms which characterize probability distributions of two or more variables. ...

Similarly for continuous random variables, the conditional probability density function can be written as p_Y|X(y | x) and this is By one convention, a random variable X is called continuous if its cumulative distribution function is continuous. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...

where p_X,Y(x, y) gives the joint distribution of X and Y, while p_X(x) gives the marginal distribution for X. Given two random variables X and Y, the joint probability distribution of X and Y is the probability distribution of X and Y together. ... Given two jointly distributed random variables X and Y, the marginal distribution of X is simply the probability distribution of X ignoring information about Y, typically calculated by summing or integrating the joint probability distribution over Y. For discrete random variables, the marginal probability mass function can be written as...

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. Borels paradox (sometimes known as the Borel-Kolmogorov paradox) is a paradox of probability theory relating to conditional probability density functions. ...

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, so that the sum over all x need not be 1. In statistics, a likelihood function is a conditional probability function considered a function of its second argument with its first argument held fixed, thus: and also any other function proportional to such a function. ...