## Probability and distributions

A probability distribution gives the frequency of occurrence of some variable. For

example, the income distribution is summarized by a function F, where F(y) gives

the fraction of the population with income at or below y. If a person is selected

at random from the population, then the probability that the person will have

income below y is F(y). Similarly, one may consider the age distribution, repre-

sented by G, so that G(a) is the fraction of people with age no larger than a, or

a wage distribution where H(w) gives the proportion of people earning a wage

no larger than w. Thus, a distribution provides information regarding the fre-

quency of a certain characteristic or property in a population. Considering the

income distribution, suppose a person is selected at random from the population

and income recorded. At the selection process, let Y be the as yet to be determined

income. This unknown number is called a random variable. The probability that

the person selected at random will have an income below y is denoted P(Y ≤ y) and this is equal to F(y). In Section 18.2, random variables are introduced and

briefly described. Intuitively, every distribution function describes the behavior

of a random variable. Section 18.3 introduces the distribution function and its

density.