CHAPTER VIII - MORE ON DISTRIBUTIONS

Abstract

Any one-dimensional distribution can be completely represented by the so-called cumulative distribution function (c.d.f.) that indicates, for each value of x, the proportion of the distributed quantity lying at points with abscissas ≤ x. Such a function P(x) is monotonic, not decreasing, starting with the value 0 and ending with the value 1. If P(x) = fxp(x) dx, one speaks of a continuous distribution and calls the derivative p(x) the density at the point x. If P(x) is discontinuous, jumping at discrete points, and remaining constant between two consecutive jumps, there is the case of a discrete distribution with lumped, concentrated values of the distributed quantity. A distribution may also be partly discrete, partly continuous. All these terms have been originally borrowed from the field of mechanics with mass or weight distributions. In addition to mean value and variance, various kinds of statistical parameters can be used to characterize a distribution V(x). The term parameter means a quantity that is uniquely determined by a distribution V(x). Two parameters derived from moments are the skewness and the kurtosis.