Various approximations for distributions are studied, especially those involving the Binomial, Poisson, gamma, and Gaussian (normal) distributions. m-procedures are used to make comparisons. A simple approximation to a continuous random variable is obtained by subdividing an interval which includes the range (the set of possible values) into small enough subintervals that the density is approximately constant over each subinterval. A point in each subinterval is selected and is assigned the probability mass in its subinterval. The combination of the selected points and the corresponding probabilities describes the distribution of an approximating simple random variable. Calculations based on this distribution approximate corresponding calculations on the continuous distribution.

In probability theory, the probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of random variables is the convolution of their corresponding probability mass functions or probability density functions respectively. Many well known distributions have simple convolutions. The following is a list of these convolutions.

This Note provides practical operational formulae to be used when one wishes to transform a probability density function of a random variable X to a random variable f(X) without affecting the underlying probability distribution. Such a process, often referred to as change of scale or transformation of coordinates, has nothing to do with the way the distribution function is displayed in a graph. This, along with the fact that the terms scale, axis and even coordinates are sometimes used interchangeably, is a frequent source of confusion.

Despite the apparent triviality of the whole matter, lengthy discussions often arise from the fact that probability density functions for f(X) are sometimes plotted in graphs with horizontal axis reporting a different function g(x). This, strictly speaking, is not illegal and, occasionally, it may be even justified by graph-appearance reasons.

Benford's law, also called the first-digit law, states that in lists of numbers from many (but not all) real-life sources of data, the leading digit is distributed in a specific, non-uniform way. According to this law, the first digit is 1 almost one third of the time, and larger digits occur as the leading digit with lower and lower frequency, to the point where 9 as a first digit occurs less than one time in twenty. This distribution of first digits arises logically whenever a set of values is distributed logarithmically. For reasons described below, real-world measurements are often distributed logarithmically (or equivalently, the logarithm of the measurements is distributed uniformly).