International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. C, ch. 8.1, pp. 679680

A probability density function, which will be abbreviated p.d.f., is a function, Φ(x), such that the probability of finding the random variable x in the interval is given by A p.d.f. has the properties and A cumulative distribution function, which will be abbreviated c.d.f., is defined by The properties of Φ(x) imply that , and Φ(x) = dΨ(x)/dx. The expected value of a function, f(x), of random variable x is defined by If f(x) = x^{n}, is the nth moment of Φ(x). The first moment, often denoted by μ, is the mean of Φ(x). The second moment about the mean, , usually denoted by σ^{2}, is the variance of . The positive square root of the variance is the standard deviation.
For a vector, x, of random variables, , the joint probability density function, or joint p.d.f., is a function, Φ_{J}(x), such that The marginal p.d.f. of an element (or a subset of elements), , is a function, , such that This is a p.d.f. for alone, irrespective of the values that may be found for any other element of x. For two random variables, x and y (either or both of which may be vectors), the conditional p.d.f. of x given y = y_{0} is defined by where is a renormalizing factor. This is a p.d.f. for x when it is known that y = y_{0}. If for all y, or, equivalently, if , the random variables x and y are said to be statistically independent.
Moments may be defined for multivariate p.d.f.s in a manner analogous to the onedimensional case. The mean is a vector defined by where the volume of integration is the entire domain of x. The variance–covariance matrix is defined by The diagonal elements of V are the variances of the marginal p.d.f.s of the elements of x, that is, . It can be shown that, if and are statistically independent, when . If two vectors of random variables, x and y, are related by a linear transformation, x = By, the means of their joint p.d.f.s are related by μ_{x} = Bμ_{y}, and their variance–covariance matrices are related by V_{x} = BV_{y}B^{T}.