Review on Probability Distributions

Transcription

1 Discrete Probability Distributios: Biomial Probability Distributio ourse: MAT, Istructor: Md. Saifuddi Khalid Review o Probability Distributios Radom Variable. A radom variable is a variable which takes specified values with specified probabilities. The probabilities are specified by the way i which the radom eperimet is coducted ad the way i which the radom variable is defied ad observed o the radom eperimet. We shall use capital letters to deote a radom variable ad the correspodig small letters to represet ay specific value of the radom variable. Probability Fuctio. If the fuctio permits us to compute the probability for ay evet that is defied i terms of value of the radom variable, the this fuctio is called a probability fuctio. Just as there are discrete ad cotiuous variables, so there are discrete ad cotiuous probability fuctios. To emphasize this distictio, we shall draw the diagram as follows: FUNTION DISRETE FUNTION ONTINUOUS FUNTION MASS FUNTION UMULATIVE MASS FUNTION DENSITY FUNTION UMULATIVE DENSITY FUNTION Discrete Probability Fuctio A probability fuctio for a discrete radom variable is called a discrete probability fuctio sice the domai of the fuctio is discrete. Probability Mass Fuctio A probability fuctio that specifies the probability that ay sigle value of discrete radom variable will occur is called a probability mass fuctio ( abbreviated as p.m.f.). If f() is the probability mass fuctio of the radom variable X, the f() = P(X = ) has the followig properties: (i) f() for all values of X; ad (ii) f () = umulative Mass Fuctio If X is a discrete radom variable with p.m.f. f(), its cumulative mass fuctio ( abbreviated as c.m.f.) specifies the probability that a observed value of X will be o greater tha. That is F() is a c.m.f. ad f() is a p.m.f., the F() =P(X ) = f ( X ). Idepedet Uiversity, Bagladesh

2 Discrete Probability Distributios: Biomial Probability Distributio ourse: MAT, Istructor: Md. Saifuddi Khalid Biomial Probability Distributio Oe widely used probability distributio of discrete radom variable is the Biomial probability distributio. It is also kow as the outcome of Beroulli process ad is associated with the ame of Jacob Beroulli. Jacob Beroulli ( -7), the first of Beroulli family of Swiss mathematicias, published a treatise o probability that cotaied the theory of permutatios ad combiatios, as well as the Biomial Theorem. A biomial eperimet or Beroulli process radom eperimet or process with the followig properties:. The eperimet cosists of a sequece of idetical trials.. Each trial has two mutually eclusive possible outcomes, such as success or failure, good or defective, yes or o, hit or miss, ad so o. The outcomes are usually called success ad failure for coveiece.. The probability of success, p, remais costat from trial to trial ( so is the probability of failure q, where q=-p.. Each trial is idepedet of other trials, i.e.,the probability of a outcome for ay particular trial is ot iflueced by the outcomes of the other trials. These coditios are satisfied if we toss a coi, say, five times. Suppose we are iterested i fidig the probability of obtaiig eactly two heads. Let us desigate head as success ad tail as failure with correspodig probabilities p ad q respectively. Suppose that oe of the sequece of outcomes of five tosses of a fair coi showig two heads is: HTHTT The probability of this specific sequece of outcome is foud by meas of a multiplicatio rule of probability ad is give by pqpqq = p q Although the resultig probability of obtaiig the specific sequece of outcomes i the order show, we are ot iterested i the order of occurrece of the successes ad failures. Rather, we are iterested i the probability of the occurrece of eactly two successes out of five tosses of a coi. I additio to the sequece show above ( call it sequece umber ) two successes ad three failures could also occur i oe of the additioal sequeces show as follows. Each of the sequeces has the same probability of occurrig, p q. Sequece Number Sequece Probability HHTTT THHTT TTHHT TTTHH HTTTH HTHTT 7 HTTHT Idepedet Uiversity, Bagladesh

3 Discrete Probability Distributios: Biomial Probability Distributio ourse: MAT, Istructor: Md. Saifuddi Khalid 8 THTTH 9 TTHTH THTHT A sigle sample of five tosses will yield oly oe sequece of successes ad failures. The questio the to be aswered is: What is the probability of gettig sequece or sequece umber or sequece umber i.e. probability of occurrig ay of these sequece umber? For fidig the aswer, the additio rule of probability is used to calculate the sum of the idividual probabilities. To get this we multiply p q by, i.e., p q. Here p =. ad q =. Therefore, the aswer is (.) (.) =.. =. As the size of the tosses icreases, it becomes more ad more difficult to list the umber of sequeces. A easy method of coutig them is required. We kow that the umber of combiatios of thigs take at a time is give by! =!( )! I our eample, =, =.! The = =.!()! Therefore, the geeral model for specifyig the probability of obtaiig eactly successes i a give umber of Beroulli trials is give by f ( ) = P[ X = ] = for =,,,, where p = the probability of a success o a sigle Beroulli trial = the umber of Beroulli trials = the umber of successes i trials This formula for the probability distributio of the umber of successes i series of Beroulli trials is called the Biomial probability distributio. It gives the probability of obtaiig eactly successes ad ( ) failures i Beroulli trials. The biomial distributio has bee etesively tabulated for differet values of ad ( see page 9 of your tet). Number of successes Probability f() = q = q p ( ) = q : : : : : = p The biomial distributio satisfies the two essetial properties of probability distributio, viz., p Idepedet Uiversity, Bagladesh

4 Discrete Probability Distributios: Biomial Probability Distributio ourse: MAT, Istructor: Md. Saifuddi Khalid (i) f() ; ad (ii) f () =. For (i), this follows from the fact that both ad p are positive ad hece q all are positive. osequetly f ( ). For (ii), we kow that biomial epasio of ( q + p) = q p Therefore, f ( ) = = ( q + p) = ( p + p) =, The biomial distributio is a family of distributios sice each differet value of or p specifies a differet distributio. I this distributio ad p are called parameters. Regardless of the value of, the distributio is symmetrical whe p=.. For small values of, whe p is greater tha., the distributio is asymmetrical, with the peak occurrig to the right of cetre, i.e., it is a egatively skewed distributio ad whe p is les tha. the distributio is asymmetrical with the peak occurrig to the left of the cetre, i.e., it is a positively skewed distributio. Mea ad Variace of Biomial Distributio The Mea. The mea of biomial radom variable, deoted by µ or E(), is the theoretical epected umber of successes i trials. µ = E( ) = = f ( ) i.e., the mea of is the sum of the products of the values that ca assume multiplied by their respective probabilities. = µ = E( ) = f ( ) = c! = =!( )! ( )! = = ( )!( )! ( )! = = ( )!( )! ( )! = = p ( )!( )! = p = p( q + p) = p [ Q ( q + p) = ] Thus the mea of biomial distributio is p. The Variace. The variace of the biomial radom variable measures the variatio of the biomial distributio ad is give by σ = E( ) µ = = f ( ) µ Here µ = p ad f ( ) = [ ( ) + ] = ( ) + = ( ) p ( q + p) + p = ( ) p + p [ Q ( q + p) = ] σ = ( ) p + p ( p) = p[( ) p + p] p, Idepedet Uiversity, Bagladesh

5 Discrete Probability Distributios: Biomial Probability Distributio ourse: MAT, Istructor: Md. Saifuddi Khalid = p [ p] = pq [Sice p+ q = ] Thus the stadard deviatio of the biomial distributio is pq ad variace = pq. Eample : The mea of biomial distributio is ad stadard deviatio. alculate,p ad q. Solutio: The mea of biomial distributio is give by p ad stadard deviatio by pq. Sice pq =; pq = ad p = Therefore, q = or q = =. 9 p = q =.9 =. Sice p =. ad q =.9 ad therefore, pq = Therefore p = or = = p Hece for give questio, =, p =. ad q =.9. Eample : Assumig that half the populatio is vegetaria so that choice of a idividual beig vegetaria is ½. Assumig that ivestigators ca take a sample of idividuals to see whether they are vegetarias, how may ivestigators would you epect to report that three people or less were vegetarias? Solutio. Probability of a perso beig a vegetaria, i.e. p = ½ q = - p = - ½ = ½ The probability that three people or less are vegetaria is give by P[X ] = P() + P() + P () + P() = q + ( ) q p + ( ) q p + ( ) q p = = = [+++] = The umber of ivestigators to report that three or less people are vegetarias is give by 7 = 7. Hece 7 ivestigators would report that or less people are vegetarias. Eample. The icidece of occupatioal disease i a idustry is such that the workers have a % chace of sufferig form it. What is the probability that out of si workers or more will cotract disease? Idepedet Uiversity, Bagladesh

6 Discrete Probability Distributios: Biomial Probability Distributio ourse: MAT, Istructor: Md. Saifuddi Khalid Solutio. The probability of a worker sufferig from disease, i.e. p = = The probability of a worker ot sufferig from disease i.e. q = p = = The probability of or more, i.e., or will cotract disease is give by P [ X ] = P[] + P[] + P[] = ( ) + ( ) + = + + = + + = [ + + ] = = =. 9 Eample. Assume that o a average oe telephoe umber out of fiftee is busy. What is the probability that if si radomly selected telephoe umbers are called (a) ot more tha three will be busy? (b) at least three of them will be busy? Solutio. p = probability that a telephoe umber is busy = q = p = = ad =. (a) The probability that out of si radomly selected telephoe umbers ot more tha three umbers are busy is give by P [ X ] = P() + P() + P() + P() = + ( ) + ( ) + ( ) [ ] = ( ) + ( ) + ( ) + ( ) = ( ) [( ) + ( ) + ( ) + ] ( ) 7 = [ ] ( ) + 87 = = (b) Probability that at least three telephoe umbers are busy is give by P [ X ] = P() + P() + P() + P() = ( ) + ( ) + ( ) + ( ) =. Idepedet Uiversity, Bagladesh

7 Discrete Probability Distributios: Biomial Probability Distributio ourse: MAT, Istructor: Md. Saifuddi Khalid Eercises from Tet, 8,,, Eercise Problem Set. The ity Bak of Durham has recetly begu a ew credit program. ustomers meetig certai credit requiremets ca obtai a credit card accepted by participatig area merchats that carries a discout. Past umbers show that percet of all applicats for this card are rejected. Give that credit acceptace or rejectio is a Beroulli process, out of applicats, what is the probability that (a) Eactly four will be rejected? (b) Eactly eight? (c) Fewer tha three? (d) More tha Five? Aswer [Ref., p. 8]: (a). (b). (c). (d).8. O average, percet of those erolled i the Federal Aviatio Admiistratio s air traffic cotroller traiig program will have to repeat the course. If the curret size at the Leesurg, Virgiia, traiig ceter is, what is the probability that (a) Fewer tha will have to repeat the course? (b) Eactly will pass the course? (c) More tha will pass the course? Aswer[Ref., p. 8]: (a).99 (b).8 (c).7. For a biomial distributio with = ad p =., use probability distributio table to fid (a) P(X = 8). (b) P(X > ) = - P(X ) (c) P(X ). Aswer [Ref., p. 7]: (a).7 (b).9 (c) Fid the mea ad stadard deviatio of the followig biomial distributios: (a) =, p =. (b) =, p =.7 (c) =, p =. (d) =, p =.9 (e) = 78, p =. Solutio [Ref.. p.9] N p µ = p σ = pq (a)...9 (b) (c)...7 (d).9.. Idepedet Uiversity, Bagladesh 7

8 Discrete Probability Distributios: Biomial Probability Distributio ourse: MAT, Istructor: Md. Saifuddi Khalid (e) The latest atiowide political poll idicates that for Americas who are radomly selected, the probability that they are coservative is., the probability that they are liberal is., ad the probability that they are middle-of-the-road is.. Assumig that these probabilities are accurate, aswer the followig questios pertaiig to a radomly chose group of Americas. (Do ot use Table.) (a) What is the probability that four are liberal? (b) What is the probability that oe are coservative? (c) What is the probability that two are middle-of-the-road? (d) What is the probability that at least eight are liberal? Solutio [Ref., p. 9]: (a) =, p =., P (X = ) = (.) (.7) =. (b) =, p =., P (X = ) = (.) (.) =. (c) 8 =, p =., P (X = ) = (.) (.8) =. 79 (d) =, p =., P (X 8) = P(X=8)+ P(X=9)+ P(X=) = ( ) ( ) ( ) ( ) ( ) ( ) 8 9 =.+.+. =.. Harley Davidso, director of quality cotrol for the Kyoto Motor compay, is coductig his mothly spot check of automatic trasmissios. I this procedure, trasmissios are removed from the pool of compoets ad are checked for maufacturig defects. Historically, oly percet of the trasmissios have such flaws. ( Assume that flaws occur idepedetly i differet trasmissios.) (a) What is the probability that Harley s sample cotais more tha two trasmissios with maufacturig flaws? (Do ot use the tables.) (b) What is the probability that oe of the selected trasmissios has ay maufacturig flaws? (Do ot use the tables.) Referece [] Richard I. Levi ad David S. Rubi, Probability Distributios, Statistics for Maagemet, Seveth Editio, p. 7, 8, [] S. P. Gupta ad M. P. Gupta, Probability Distributios, Busiess Statistics, Twelfth Revised ad Elarged Editio,, ISBN , pp. 8-9 [] David R. Aderso, Deis J. Sweeey, Thomas A. Williams, Discrete Probability Distributios, Statistics for Busiess ad Ecoomics, Eighth Editio, pp Idepedet Uiversity, Bagladesh 8