1. Characteristics of Probability Distributions

Statistics deals with random processes. The outcomes of such processes, for example, the throwing of a die or the number of disintegrations in a particular radioactive source in a period of time T, fluctuate from trial to trial such that it is impossible to predict with certainty what the result will be for any given trial. Random processes are described, instead, by a probability density function which gives the expected frequency of occurrence for each possible outcome. More formally, the outcome of a random process is represented by a random variable x, which ranges over all admissible values in the process. If the process is the throwing of a single die, for instance, then x may take on the integer values 1 to 6. Assuming the die is true, the probability of an outcome x is then given by the density function P(x) = 1/6, which in this case happens to be the same for all x. The random variable x is then said to be distributed as P(x).

Depending on the process, a random variable may be continuous or discrete. In the first case, it may take on a continuous range of values, while in the second only a finite or denumerably infinite number of values is allowed. If x is discrete, P(x_i) then gives the frequency at each point x_i. If x is continuous, however, this interpretation is not possible and only probabilities of finding x in finite intervals have meaning. The distribution P(x) is then a continuous density such that the probability of finding x between the interval x and x + dx is P(x)dx.