International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 8.1, pp. 679-680

## Section 8.1.1.2. Statistics

E. Princea and P. T. Boggsb

aNIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA, and bScientific Computing Division, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA

#### 8.1.1.2. Statistics

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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) = xn, 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 σ2, 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, ΦJ(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 = y0 is defined by where is a renormalizing factor. This is a p.d.f. for x when it is known that y = y0. 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 one-dimensional 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 μx = Bμy, and their variance–covariance matrices are related by Vx = BVyBT.