Find an application of one of the special distributions that we have learned in

ID: 3152181 • Letter: F

Question

Find an application of one of the special distributions that we have learned in chapter 3 and 4 related to your discipline or in the news. You may find such applications in material you've learned from another class, in a research article, through your job, in a news/magazine article, etc. The application should just be related in some way to the concepts we've learned. The discrete distributions are the binomial, hypergeometric, negative binomial, and Poisson. The continuous distributions are the uniform, normal, and exponential. There are some other distributions given in the book that we did not discuss in detail (e. g., Chi-square, gamma, Weibull distributions) and you are encouraged to investigate one of these if you choose. Guidelines: Provide a concise, well written summary of the application. The following are some guidelines for items to include in the summary: Clearly state the distribution from chapter 3 or 4 that you will be discussing. State the main goals of the application and the questions that are being addressed. State how the distribution is used to help address the goals of the problem. If applicable, discuss the main results. If applicable, cite the reference(s) you used. Please type your summary in Times New Roman, 12pt font. Length should be approximately 1 page single spaced with 1" margins.

Explanation / Answer

Distribution

Application

Example

Comment

Binomial

Gives probability of exactly x successes in n independent trials, when probability of success p on single trial is a constant. Used frequently in quality control, reliability, survey sampling and other industrial problem.

Can sometimes be approximated by normal or by Poisson distribution.

Hyper

geometric

Gives probability of picking exactly x good units in a sample of n units from a population of N unit when there are k bad units in the population. Used in quality control and related applications.

Given a lot with 21 good units and four defectives. What is the probability that a sample of five will yield mot more than one defective?

May be approximated by binomial distribution when n is small relative to N.

Negative Binomial

Gives probability similar to Poisson distribution when events do not occur at a constant rate and occurrence rate is a random variable that follows a gamma distribution.

Distribution of number of cavities for a group of dental patients.

Generalization of Pascal distribution when s is not an integer. Many authors do not distinguish between Pascal and negative binomial distribution.

Poisson

Gives probability of exactly x independent occurrence during a given period of time if event takes place independently and at a constant rate. May also represent number of occurrences over constant areas or volumes. Used frequently quality control, reliability, queuing theory, and so on.

Used to represent distribution of material, customer arrivals, insurance claims, incoming telephone calls, alpha particles emitted, and so on.

Frequently used as approximation to binomial distribution.

Uniform

Gives probability that observation will occur within a particular interval when probability of occurrence within that interval is directly proportional to interval length.

Used to generate random values.

Special cases of beta distribution.

Normal

A basic distribution of statistics. Many applications arise from central limit theorem. Consequently, appropriate model for many but not all physical phenomena.

Distribution of physical measurements on living organisms, intelligence tests scores, product dimensions, average temperatures, and so on.

Tabulation of cumulative values of standardized normal distribution readily available. Many methods of statistical analysis presume normal distribution.

Exponential

Gives distribution of time between independent events at a constant rate. Equivalently, probability distribution of life, presuming constant conditional failure rate. Consequently, application in many but not all reliability situation.

Distribution of time between arrivals of particles at a counter. Also life distributions of complex non-redundant systems, and usage life of some components.

Special case of both Gamma and Weibull distribution.

Gamma

A basic distribution of statistics for variables bounded at one side for example, 0x<. Gives distribution of time required for exactly k independent events to occur, assuming events take at a constant rate. Used frequently in queuing theory, reliability, and other industrial applications.

Distribution of time between recalibrations of instrument that needs recalibration after k uses; time between inventory restocking, time to failure for a system with stand by components.

Cumulative distribution values have been tabulated. Exponential and chi-square distribution are special cases.

Weibull

General time-to-failure distribution due to wide diversity of hazard rate curves, and extreme-value distribution for minimum of N values from distribution bounded at left.

Life distribution for some capacitors, ball bearings, relays, and so on.

Rayleigh and exponential distributions are special cases.