Continuous Distributions

Abstract

A variable, x, is continuous when x can be any number between two limits. For example, a scale measures a boy at 150 pounds; and assuming the scale is correct within one-half pound, the boy’s actual weight is a continuous variable that could fall anywhere from 149.5 to 150.5 pounds. The variable, x, is a continuous random variable when a mathematical function, called the probability density defines the shape along the admissible range. The density is always zero or larger and the positive area below the density equals one. Each unique continuous random variable is defined by a probability density that flows over the admissible range. Eight of the common continuous distributions are described in the chapter. For each of these, the range of the variable is stated, along with the probability density, and the associated parameters. Also described is the cumulative probability distribution that is needed by an analyst to measure the probability of the x falling in a sub-range of the admissible region. Some of the distributions do not have closed-form solutions, and thereby, quantitative methods are needed to measure the cumulative probability. Sample data is used to estimate the parameter values. Examples are included to demonstrate the features and use of each distribution. The distributions described in this chapter are the following: continuous uniform, exponential, Erlang, gamma, beta, Weibull, normal and lognormal. The continuous uniform occurs when all values between limits a to b are equally likely. The normal density is symmetrical and bell shaped. The exponential happens when the most likely value is at x = 0, and the density tails down in a relative way as x increases. The density of the Erlang has many shapes that range between the exponential and the normal. The shape of the gamma density varies from exponential-like to one where the mode (most likely) and the density skews to the right. The beta has many shapes: uniform, ramp down, ramp up, bathtub-like, normal-like, and all shapes that skew to the right and in the same manner they skew to the left. The Weibull density varies from exponential-like to shapes that skew to the right. The lognormal density peaks near zero and skews far to the right.