Continuous distributions. Density functions

14.9 Continuous distributions. Density functions

be a one-dimensional random variable and let F be its distribution function, so that F(t) =

Let

for every real If the probability = is zero for every then, because of Theorem 14.5, F is continuous everywhere on the real axis. In this case F is called a continuous distribution and

is called a continuous random variable. If the deriva- tive F’ exists and is continuous on an interval [a, t] we can use the second fundamental theorem of calculus to write

F(a) is, of course, the probability <

where f is the derivative of F. The difference F(t)

and Equation (14.13) expresses this probability as an integral. Sometimes the distribution function F can be expressed as an integral of the form in which the integrand f is integrable but not necessarily continuous. Whenever an equation such as (14.13) holds for all intervals [a,

a proba- bility density function of the random variable

the integrand f is called

(or of the distribution F) provided thatfis nonnegative. In other words, we have the following definition:

DEFINITION OF A PROBABILITY DENSITY FUNCTION . Let X be a one-dimensional random variable with a continuous distribution function F. A nonnegative function f is called a probability density of (or of F)

is integrable on every interval [a, t] and (14.14)

F(t) F(a) =

du.

0 and we obtain the important formula (14.15)

If we let a in (14.14) then F(a)

F(t) =

t) =

f(u) du ,

valid for all real If we now let

and remember that F(t)

1 we find that

Calculus of probabilities

For discrete random variables the sum of all the probabilities = is equal to 1. Formula (14.16) is the continuous analog of this statement There is also a strong analogy between formulas (14.11) and (14.15). The density functionf plays the same role for con-

tinuous distributions that the probability mass plays for discrete distributions integration takes the place of summation in the computation of probabilities. There is one important difference, however. In the discrete case p(t) is the probability that

= but in the continuous

= . In fact, this probability is zero because F is continuous for every

is

the probability that

Of course, this also means that for a continuous distribution we have

If F has a density of these probabilities is equal to the integral f(u) du.

Note:

A given distribution can have more than one density since the value of the

integrand in (14.14) can be changed at a finite number of points without altering the integral. But iffis

in this case the value of the density function at is uniquely determined by

at

= F’(t)

Since is nonnegative, the right-hand member of Equation (14.14) can be interpreted geometrically as the area of that portion of the ordinate set

to the left of the line x = . The area of the entire ordinate set is equal to 1. The area of the portion of the ordinate set above a given interval (whether it is open, closed, or half-open) is the proba- bility that the random variable

takes on a value in that interval. Figure 14.6 shows an

example of a continuous distribution function F and its density function f. The ordinate

F(t) in Figure 14.6(a) is equal to the area of the shaded region in Figure 14.6(b).

The next few sections describe some important examples of continuous distributions.