# Introduction to Analytical Probability Distributions

15 Pages Posted: 1 Jun 2017

See all articles by Robert L. Carraway

## Robert L. Carraway

University of Virginia - Darden School of Business

## Robert Jenkins

University of Virginia - Darden School of Business

### Abstract

This technical note provides a mostly nontechnical introduction to analytical probability distributions. The distributions covered are: uniform, triangular, normal, Poisson, exponential, lognormal, and binomial.

Excerpt

UVA-QA-0690

March 29, 2010

Introduction to Analytical Probability Distributions

Imagine that you are a U.S.-based manufacturer of components for consumer electronics. You are very concerned about product quality in today's highly competitive market. During your latest review of quality information, you were troubled by data regarding one of your manufacturing machines—we'll call it Machine A.

In the latest production run, Machine A produced 10 components. Two of those ten components were defective. You have been reviewing quality data for a long time; you know that, on average, only 5% of all the components produced by any given machine are defective. Thus, you are worried that there might be something wrong with Machine A. On the other hand, the 20% defect rate in the latest production run might be just a fluke. So the question is: Given that Machine A produced 2 defects out of 10 products in the latest production run, and given that the average machine would only produce .5 defects out of 10 products in the average run, do you think Machine A is itself “defective”?

You can use a number of statistical tools to help answer this question, most of which compare probability distributions. In the above example, you know how often a well-functioning machine produces a defective product (1 defect for every 20 products). You know this because you have observed the behavior of a large number of machines, all of which you know to be functioning properly, over a large number of production runs. When combined, these observations comprise a “population”—in this case, the population of properly functioning machines. This population generates a defect probability distribution in which, on average, 1 out of every 20 products is defective. Of course, this 5% defect rate is only an average figure—some machines in this population will operate at a higher defect rate (e.g., 1 defect for every 10 products), and some machines will operate at a lower defect rate (e.g., 1 for every 50 products).

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Keywords: statistics, probability, distributions, decision analysis, risk analysis, simulation, forecasting

Suggested Citation

Carraway, Robert L. and Jenkins, Robert, Introduction to Analytical Probability Distributions. Darden Case No. UVA-QA-0690, Available at SSRN: https://ssrn.com/abstract=2975104