Continuous random variables are variables that take values that could be any
real number within some interval. One common example of such variables is
*time*, for example, the time to failure of a system or the time to
complete some task. Other examples include physical measurements such as
length or diameter. As will be seen, continuous random variables also can be
used to approximate discrete random variables.

To develop probability models for continuous r.v.'s, it is necessary to make
one important restriction: we only consider events associated with these r.v.'s
that are defined in terms of intervals of real numbers, including intersections
and unions of intervals. Probability models are constructed by representing
the probability that a r.v. is contained within an interval as the area under
a curve over that interval. That curve is called the *density function*
of the r.v. To satisfy the laws of probability, density functions must satisfy
the following two conditions:

The probability that the r.v. is contained within an interval is then

Note that in the case of continuous r.v.'s,

since the area under a curve at a point is 0. The distribution function of a continuous r.v. is given by

Note that the Fundamental Theorem of Calculus implies that

Also note that the value of a density function is not a probability; nor is a density necessarily bounded by 1. It can be thought of as the concentration of likelihood at a point.

The expected value of a continuous r.v. is defined analogously to the expected
value of a discrete r.v. with the p.m.f. replaced by the density function and
the sum replaced by an integral:

Also, the variance of a continuous r.v. is defined by

where . Note that the additive property of integrals gives

where .

To construct probability models for continuous r.v.'s, it is only necessary to find a density function that models appropriately the concentration of likelihood.

2018-10-18