**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