 in probability theory and statistics, 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; in some cases the conditional probabilities may be expressed as functions containing the unspecified value x of x as a parameter. in case that both "x" and "y" are categorical variables, a conditional probability table is typically used to represent the conditional probability. the conditional distribution contrasts with the marginal distribution of a random variable, which is its distribution without reference to the value of the other variable. if the conditional distribution of y given x is a continuous distribution, then its probability density function is known as the conditional density function. the properties of a conditional distribution, such as the moments, are often referred to by corresponding names such as the conditional mean and conditional variance. more generally, one can refer to the conditional distribution of a subset of a set of more than two variables; this conditional distribution is contingent on the values of all the remaining variables, and if more than one variable is included in the subset then this conditional distribution is the conditional joint distribution of the included variables. ==discrete distributions== for discrete random variables, the conditional probability mass function of y given the occurrence of the value x of x can be written according to its definition as: :p_y(y\mid x = x)=p(y = y \mid x = x) = \frac{p(x=x\ \cap y=y)}{p(x=x)}. due to the occurrence of p(x=x) in a denominator, this is defined only for non-zero (hence strictly positive) p(x=x). the relation with the probability distribution of x given y is: :p(y=y \mid x=x) p(x=x) = p(x=x\ \cap y=y) = p(x=x \mid y=y)p(y=y). ==continuous distributions== 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 :f_y(y \mid x=x) = \frac{f_{x, y}(x, y)}{f_x(x)}, where fx,y(x, y) gives the joint density of x and y, while fx(x) gives the marginal density for x. also in this case it is necessary that f_x(x)>0. the relation with the probability distribution of x given y is given by: :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. ==relation to independence== random variables x, 
Probability_theory Types_of_probability_distributions