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
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:
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.