Continuous Random Variables

Summary :

A continuous random variable is measured, not counted, so probability is found as area under a probability density function, not at single points. This chapter defines continuous probability functions, then develops two models: the uniform distribution, where outcomes are equally likely, and the exponential distribution, used for waiting times and decay.

Probability as Area Under a Curve

A continuous random variable is measured rather than counted, so its values fall across an interval instead of at separate points. Its behaviour is described by a probability density function, written f(x), whose graph is a curve. Probability is represented by the area between that curve and the x-axis, and the entire area under the curve equals one. Because probability is area, it is defined for intervals of x rather than for any single value, so the probability that x equals one exact point is zero.

The Uniform Distribution

In a uniform distribution every outcome in the range is equally likely, and the density function is a horizontal line of constant height. For a variable spread evenly between a and b, f(x) equals 1/(b − a) across that interval, and probabilities are found as the area of a rectangle: base times height. For example, with f(x) = 1/20 on the interval 0 to 20, the probability that x lies between 0 and 2 is (2 − 0)(1/20) = 0.1. The distribution is written X ~ U(a, b).

The Exponential Distribution

The exponential distribution models the time between events, such as how long a phone call lasts or a car battery survives, and is written X ~ Exp(m), where m is the decay parameter. Its density function is f(x) = m·e^(−m·x), and the cumulative distribution function P(X < x) = 1 − e^(−m·x) gives probability as area up to a point. Unlike the uniform distribution, smaller values are more likely than larger ones, so the curve falls away from left to right. Worked examples in the chapter use it to find warranty periods and expected lifetimes.