Other Distributions

There are many other distributions that are useful in operations, sales, engineering, etc. They are amply described in the literature, ^[12] and a brief listing is given in Table 2-3.

Table 2-3: Other Distributions
Distribution	General Application
Poisson	The Poisson distribution is used for counting the random arrival or occurrence of an event in a given time, area, distance, etc. For example, the random clicks of a Geiger counter or the random arrival of customers to a store or website is generally Poisson distributed. The Poisson distribution has a parameter, λ for arrival rate. As λ becomes large, the Poisson distribution is approximately Normal with μ = λ.
Binomial	The Binomial distribution applies to events that have two outcomes, like the coin toss, where the outcomes are generally referred to as success or failure, true or false. If X is the number of successes, n, in a series of repeated trials, then X will have a Binomial distribution. As n becomes large, the Binomial distribution is approximately Normal. The number of heads in a coin toss is Binomial for small n, becoming all but Normal for large n.
Rayleigh	The Rayleigh distribution is an asymmetrical distribution of all positive outcomes. It approximates outcomes that tend to cluster around a most likely value, but nonetheless have a finite probability of great pessimism. The Rayleigh has a single parameter, "b", that is the most likely value of the outcome.
Student's t	The Student's t, or sometimes just t-distribution, is used in estimating confidence intervals when the variance of the population is unknown and the sample size is small. The Student's t is closely related to the Normal distribution, being derived from it. The Student's t has a parameter v, for "degrees of freedom." For large values of v, the Student's t is all but Normal.
Chi-square	The distribution of random variables of the form χ² is often the Chi-square, named after the Greek letter chi χ. The Chi-square distribution is always positive and highly asymmetrical, appearing like a decaying exponential when a parameter, n, for degrees of freedom, is small. The Chi-square finds application in hypothesis testing and in determining the distribution of the sample variance.

^[9]The probability function is also known as the "distribution function" or "probability distribution function."

^[10]If the sum of "a" and "b" is a large number, then the BETA will be more narrow and peaked than a Normal; the ratio of a/b controls the asymmetry of the BETA.

^[11]Carl Friedrich Gauss (1777-1855) in 1801 published his major mathematical work, Disquisitiones Arithmeticae. Gauss was a theorist, an observer, astronomer, mathematician, and physicist.

^[12]Downing, Douglas and Clark, Jeffery, Statistics the Easy Way, Barrons Educational Series, Hauppauge, NY, 1997, pp. 90–155.