Important Probability Distributions

1 Important Probability DistributionsOPRE 6301 Important Distributions ..Certain Probability Distributions occur with such regular-ity in real-life applications that they have been given theirown names. Here, we survey and study basic propertiesof some of will discuss the following Distributions : Binomial Poisson Uniform Normal ExponentialThe first two are discrete and the last three Distribution..Consider the following scenarios: The number of heads/tails in a sequence of coin flips Vote counts for two different candidates in an election The number of male/female employees in a company The number of accounts that are in compliance or notin compliance with an accounting procedure The number of successful sales calls The number of defective products in a production run The number of days in a month your company s com-puter network experiences a problemAll of these are situations where the binomial distributionmay be Framework.

2 There is a set of assumptions which, if valid, would leadto a binomial distribution. These are: A set ofnexperiments ortrialsare conducted. Each trial could result in either asuccessor afailure. The probabilitypof success is thesamefor all trials. The outcomes of different trials areindependent. We are interested in the total number of successes the above assumptions, letXbe the total numberof successes. Then,Xis called abinomial randomvariable, and the Probability distribution ofXis calledthebinomial Probability -Mass Function..LetXbe a binomial random variable . Then, its Probability -mass function is:P(X=x) =n!

3 X!(n x)!px(1 p)n x(1)forx= 0, 1, 2, .. , values ofnandpare called theparametersof understand (1), note that: The Probability for observinganysequence ofnin-dependent trials that containsxsuccesses andn xfailures ispn(1 p)n x. The total number of such sequences is equal to(nx) n!x!(n x)!( , the total number of possible combinations whenwe randomly selectxobjects out ofnobjects).4 Example: Multiple-Choice ExamConsider an exam that contains 10 multiple-choice ques-tions with 4 possible choices for each question, onlyone of which is a student is to select the answer for every ques-tion randomly.

4 LetXbe the number of questions thestudent answers correctly. Then,Xhas a binomialdistribution with parametersn= 10 andp= (Convince yourself that all assumptions for a binomialdistribution are reasonable in this setting.)What is the Probability for the student to get no answercorrect? Answer:P(X= 0) =10!0!(10 0)!( )0(1 )10 0= ( )10= is the Probability for the student to get two an-swers correct? Answer:P(X= 2) =10!2!8!( )2(1 )8= 45 ( )2 ( )8= is the Probability for the student to fail the test( , to have less than 6 correct answers)? Answer:P(X 5) =5 i=0P(X=i)= + + + + + probabilities can be computed using the Excelfunction BINOMDIST().

5 Two other examples are givenin a separate Excel Mean and Variance..It can be shown that =E(X) =npand 2=V(X) =np(1 p) .For the previous example, we have E(X) = 10 = V(X) = 10 ( ) (1 ) = Distribution..The Poisson distribution is another family of distributionsthat arises in a great number of business situations. Itusually is applicable in situations where random events occur at a certainrateover a period the following scenarios: The hourly number of customers arriving at a bank The daily number of accidents on a particular stretchof highway The hourly number of accesses to a particular webserver The daily number of emergency calls in Dallas The number of typos in a book The monthly number of employees who had an ab-sence in a large company Monthly demands for a particular productAll of these are situations where the Poisson distributionmay be Framework.

6 Like the Binomial distribution, the Poisson distributionarises when a set of canonical assumptions are reasonablyvalid. These are: The number of events that occur in any time intervalis independent of the number of events in any otherdisjoint interval. Here, time interval is the standardexample of an exposure variable and other interpre-tations are possible. Example: Error rate perpageina book. The distribution of number of events in an interval isthe same for all intervals of the same size. For a small time interval, the Probability of observ-ing an event is proportional to the length of the inter-val.

7 The proportionality constant corresponds to the rate at which events occur. The Probability of observing two or more events inan interval approaches zero as the interval the above assumptions, let be the rate at whichevents occur,tbe the length of a time interval, andXbethe total number of events in that time interval. Then,Xis called aPoisson random variableand the proba-bility distribution ofXis called thePoisson t; then, can be interpreted as the average, ormean, number of events in an interval of Probability -Mass Function..LetXbe a Poisson random variable . Then, its Probability -mass function is:P(X=x) =e xx!

8 (2)forx= 0, 1, 2, ..The value of is theparameterof the distribution. Fora given time interval of interest, in an application, canbe specified as times the length of that : TyposThe number of typographical errors in a big textbookis Poisson distributed with a mean of per 100 pages of the book are randomly is the Probability that there are no typos? An-swer:P(X= 0) =e xx!=e != 400 pages of the book are randomly are the probabilities for having no typos andfor having five or fewer typos? Answers:P(X= 0) =e 4( 4)00!= (X 5) =5 i=0P(X=i)= + + + + + probabilities can be computed using the Excelfunction POISSON().

9 Further numerical examples of thePoisson distribution are given in a separate Excel and VarianceIt can be shown thatE(X) = andV(X) = .Interpretation of (2)The form of (2) seems mysterious. The best way to un-derstand it is via the binomial a time interval and divide it intonequally-sizedsubintervals. Supposenis very large so that either oneor zero event can occur in a subinterval. Suppose furtherthat the Probability for an event to occur in a subintervalis /n, independent of what occurs in other these assumptions, the total number of events,X,in that interval has a binomial distribution with parame-tersnand /n.

10 That is,P(X=x) =n!x!(n x)!( n)x(1 n)n x(3)forx= 0, 1, 2, .. , thatE(X) =n ( /n) = , suggesting that (3) and(1) are consistent. Indeed, it can be shown that asnapproaches , (3) becomes (2). This useful fact is calledPoisson approximationto the binomial will see several other examples of such limiting ap-proximations in future chapters. They provide simpleand accurate approximations to otherwise Continuous Distributions ..Recall that a continuous random variable or distribu-tion is defined via a probabilitydensityfunction. Letf(x) (nonnegative) be the density function of ,f(x) is the rate at which Probability accumulatesin the neighborhood ofx.