ENGI 4421 Discrete Probability Distributions Page Discrete Probability Distributions [Navidi sections ; Devore sections

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1 ENGI 441 Discrete Probability Distributios Page 9-01 Discrete Probability Distributios [Navidi sectios ; Devore sectios ] Chater 5 itroduced the cocet of robability mass fuctios for discrete radom quatities. The oly stadard distributio itroduced the is the discrete uiform distributio, whose.m.f. is 1 1,,, [ad zero otherwise] This chater will itroduce some more stadard robability mass fuctios. Beroulli Probability Distributio Where the outcome of a sigle trial ca take oe of oly two ossible values, success or failure, the Beroulli radom quatity mas success to X = 1 ad failure to X = 0. Success ad failure are a comlemetary air of simle evets. Success ca mea heads o oe fli of a coi, or defective whe a sigle item is tested, or total is 7 o oe roll of two dice, etc. If the robability of a success is, the the.m.f. follows immediately: The mea ad variace are calculated easily: 1 X E 0 1 ad E i 0 1 i X otherwise X X X V E E 1

2 ENGI 441 Discrete Probability Distributios Page 9-0 Eamle 9.01: Let X = 1 corresod to total score o oe roll of a air of fair stadard si-sided dice is 7 ad X = 0 i the evet of some other total, the X has a Beroulli distributio. Fid the arameter of this distributio ad fid the mea ad variace of X. There are 36 equally likely ossible airs of scores. Si of them roduce a total of 7. Therefore EX ad VX The Biomial Distributio The most imortat discrete robability distributio is the biomial distributio. develo the robability mass fuctio through a eamle. Here we Eamle 9.0 It is kow that each comoet, as it emerges from a eerimetal roductio rocess, has a robability of 0% to be defective (ad therefore a robability of 80% to be good). A radom samle of te comoets is draw. (a) Fid the robability that eactly three comoets i the radom samle are defective. I a sigle trial (a sigle comoet i the radom samle) let S = success (the comoet is defective) ad F = failure (the comoet is good) ( = S ~ ) [It may seem strage to associate success with defective, but it is the umber of defective items i which we are iterested.] This is a Beroulli trial (a air of comlemetary outcomes oly), with costat P[S]. Also let E = the desired evet (three defective ad seve good comoets i the radom samle) The oe way i which the desired evet ca occur is if the first three comoets tested are all defective ad the remaiig seve are all good. The robability of this outcome (SSSFFFFFFF i that order) is S F P

3 ENGI 441 Discrete Probability Distributios Page 9-03 Eamle 9.0(a) (cotiued) However, iside the evet E, the three successes may occur aywhere amog the te trials. The umber of distict rearragemets of three successes i te trials is 10 C Therefore P[E] = (b) Fid the robability mass fuctio for X = (the umber of defective comoets i the radom samle). P 10 S F But these successes ca be rearraged amog the 10 trials i 10 C distict ways. Therefore the robability mass fuctio of X is P[eactly successes] = P[X = ] =.0.80 C ( = 0, 1,,..., 10) We ca geeralize this result further. Let the roortio of defective items i the oulatio be (rather tha 0%) ad let the samle size be (rather tha 10). The the biomial robability mass fuctio becomes P ;, 1 X b I this eamle, we have draw a radom samle from a cocetual oulatio, so that every trial is ideedet of every other trial.

4 ENGI 441 Discrete Probability Distributios Page 9-04 Coditios for a discrete radom quatity to have a biomial.m.f.: I geeral, if the radom quatity X reresets the umber of successes i trials the X will have a biomial robability distributio if ad oly if the followig coditios all hold: (1) Each trial has eactly two comlemetary outcomes ( success ad failure ); () The [ucoditioal] robability of success is costat across all trials; (3) The outcome of each trial is ideedet of all other trials; (4) The samle size is fied. Coditio (3) holds oly if the samlig is either with relacemet or from a ifiite oulatio (or both). If the samlig is without relacemet from a fiite but large oulatio, the coditio (3) may be aroimately true. Eamle 9.03 Show that coditio (3) is ot satisfied if a radom samle of size is take from a oulatio of size 5 with 40% successes. Show that coditio (3) is almost satisfied if a radom samle of size is take from a oulatio of size 5000 with 40% successes. S S F F F Let S 1 = success o trial 1 ad S = success o trial the P S but P S S Therefore the outcomes of the two trials are ot ideedet. [The eact robability distributio is hyergeometric.],000 S s 3,000 F s Let S 1 = success o trial 1 ad S = success o trial 000 the P S but P S S1 1, ,999 Therefore the outcomes of the two trials are early ideedet ad the biomial.m.f. may be used as a ecellet aroimatio to the true robability distributio.

5 ENGI 441 Discrete Probability Distributios Page 9-05 The biomial cumulative distributio fuctio is y B ;, b ;, y 0 = b(0;, ) + b(1;, ) + b(;, ) b(;, ) The biomial cdf B(;, ) is tabulated i Navidi (ad Devore), table A.1, for u to 0 ad values of that are iteger multiles of.05. It ca be evaluated for ay valid choice of (, ) usig the file at " Eamle 9.04 The robability mass fuctio of the radom quatity X is kow to be biomial with arameters = 10 ad =.0. Fid P[1 < X < 5]. P[1 < X < 5] = P[1 < X 4] = B(4; 10,.0) B(1; 10,.0) = (usig Table A.1) =.591 or, usig the Web file "Biomial.ls", B(4; 10,.60) B(1; 10,.60) = OR P[1 < X < 5] = P[X = ] + P[X = 3] + P[X = 4] C C3 C4 = = Recall that P a X b F b F a F where the c.d.f. is

6 ENGI 441 Discrete Probability Distributios Page 9-06 Eamle 9.04 (cotiued) A sreadsheet ca be used to carry out these calculatios. MINITAB ad Ecel both cotai the biomial mf ad cdf, as does the Ecel file at " To fid the values of B(4; 10,.0) ad B(1; 10,.0), the followig artial table of values for b(; 10,.0) ad B(; 10,.0), (draw from the Ecel file), may be used: = 10 = 0..m.f. c.d.f. P[X = ] P[X ] If a questio ivolvig the calculatio of several adjacet biomial robability masses arises i a test or eamiatio, the the aroriate table of values, similar to that show above, will be rovided with the questio aer. [Oe could also just add u the values of the.m.f. for, 3, 4 to fid P1 X 5 ]

7 ENGI 441 Discrete Probability Distributios Page 9-07 Eamle 9.05 Te er cet of all items i a large roductio ru are kow to be defective. A radom samle of 0 items is draw. (a) (b) (c) Prove that the radom quatity X = (the umber of defective items i the radom samle) has a biomial robability mass fuctio. Fid the robability that more tha two items i the radom samle are defective. How may defective items does oe eect to fid i the radom samle? (a) Let X = the umber of defective items i the radom samle. each trial (item) has a comlemetary air of outcomes (defective = success, good = failure ) P[success] = costat = 10% Trials are ideedet to a good aroimatio (because the radom samle is draw from a large oulatio) = 0 is fied. All four coditios are satisfied. Therefore X does follow a biomial robability distributio (at least to a ecellet aroimatio). P[X = ] = b(; 0,.10). (b) P[X > ] = 1 P[X ] = 1 B(; 0,.10) = OR (by direct calculatio of the.m.f. values), 1 P[X ] = 1 (P[X = 0] + P[X = 1] + P[X = ]) = 1 { (.9) 0 + 0(.9) 19 (.1) (.9) 18 (.1) } = 1 ( ) Side ote: P[X > ] = P[ < X 0] = B(0; 0,.1) B(; 0,.1) But B(;, ) = 1 for ay : it absolutely certai to get at most successes i attemts. Therefore P[X > ] = 1 B(; 0,.1) (as i the mai solutio)

8 ENGI 441 Discrete Probability Distributios Page 9-08 Eamle 9.05 (cotiued) To fid the value of B(; 0,.10), the artial table of values for b(; 0,.10) ad B(; 0,.10) may be used: = 0 = 0.1.m.f. c.d.f. P[X = ] P[X ] (c) For ay biomial radom quatity X, Thus = E[X] = 0.10 = X X E ad V 1 [Additioal Notes: Oe eects to see the same roortio of successes i a samle as i the oulatio. The outcome of a sigle trial is a Beroulli radom quatity X i, for which the variace is V X i 1 (see age 9.01). The biomial radom quatity X is the sum of of these Beroulli radom quatities, so that VX V Xi V Xi (because the radom quatities are ideedet) i i V X i A more rigorous derivatio is o the et age.] i

9 ENGI 441 Discrete Probability Distributios Page 9-09 The roof of the formulae for the mea ad variace of the biomial radom quatity is ot eamiable, but is reseted here: 0 0! E[ X ] b( ; ; ) (1 )!( )! 1 1! ! 1 1! 1 1 Let y = 1 ad m = 1 the m m m! y my E[ X ] 1 b y; m, 1 y!( m y)! = y 0 y 0! X!! E b ;, !! !! 1!!!! 0 1 1!! 1!! 1 But the secod of these summatios is, from the derivatio of E[X] above, equal to E[X].! E X 1 1!! Let y = ad m = the m! y E 1 1 y! m y! m my X y 0 m 1 b ;, 1 y m y 0 Therefore VX E X EX 1

10 ENGI 441 Discrete Probability Distributios Page 9-10 Eamle 9.06 Fid the robability of (a) eactly oe 5 or 6 whe four dice are throw. (b) at least oe 5 or 6 whe four dice are throw. (c) How may times, o average, do you eect a 5 or 6 to occur whe four dice are throw? Let X = (umber of times a 5 or 6 occurs), the P[ X = ] = b(; 4, 1/3). [Oe ca check that all four coditios for a.m.f. to be biomial are valid.] (a) P[X = 1] = b(1; 4, 1/3) = 1 C C (b) P[X 1 ] = 1 P[X < 1] = 1 P[X = 0] = (c) E[X] =

11 ENGI 441 Discrete Probability Distributios Page 9-11 Eamle 9.07 A roductio rocess i a factory has a defect rate of %. What is the smallest samle size for which the robability of ecouterig at least oe defective item eceeds 95%? I other words, fid the least such that P[X > 0] > 0.95, whe P[X = ] = b(;,.0). P[X > 0] = 1PX 0 1 P X But we require PX l 1.95 l.98 ['l' is a mootoic fuctio] l.05 l.98 l.98 l.05 l(.05).9957 [sig reverses egative divisor] l(.98) mi 149 [ot 148, as ca be see below.] As a check, = 148 P[X > 0] = 1 (.98) % = 149 P[X > 0] = 1 (.98) %.

12 ENGI 441 Discrete Probability Distributios Page 9-1 Estimatio of Poulatio Proortio I may cases we do ot kow the value of the success robability (which is also the roortio of successes i the etire oulatio). A oit estimate of a ukow value of may be foud by coductig ideedet Beroulli trials that is, take a radom samle of size from the oulatio ad for each item record whether it is a success or a failure. The samle roortio P is a radom quatity that is a estimator for : P umber of successes X umber of trials A actual observed samle roortio is a estimate for :. The radom quatity X is biomial with arameters (, ) E P E X 1 EX 1 Therefore the samle roortio P is a ubiased estimator of the oulatio roortio. V P X 1 1 V VX 1 1 The ucertaity i P is 1 Sice is the very arameter that we are tryig to estimate, it ad are ukow. The simlest solutio is to relace the ukow by its oit estimate i the ˆ 1 ˆ eressio for the ucertaity: [Later we shall see a better estimate.] Eamle 9.08 Suose that there are 13 successes i a radom samle of size 0 draw from a much larger oulatio. The our estimate of the true roortio of successes i the oulatio is ˆ ˆ ˆ

13 ENGI 441 Discrete Probability Distributios Page 9-13 Hyergeometric Probability Distributio This distributio will also be develoed by way of a eamle. Eamle 9.09 A truck carries ie crates of riter aer ad three crates of rig biders. Four crates are take at radom from the truck. Fid the robability that at least two of the four crates cotai rig biders. Let X = the umber of crates of rig biders i the radom samle of four crates. P X is required. Let us check the four coditios for a biomial distributio: Comlemetary air of outcomes i each trial: Each crate either cotais rig biders (= success ) or riter aer (= failure ) P[S] = costat For each crate i the samle, P[S] = 3/1 (whe you do t kow the cotets of the other crates). Trials are ideedet 3 PS 0.5 but P S S PS 1 11 This coditio therefore fails (ad is ot eve close to beig true). Samle size is costat = 4 Therefore the biomial.m.f. caot be used (ot eve as a aroimatio). Eamie the oulatio ad the samle (et age):

14 ENGI 441 Discrete Probability Distributios Page 9-14 Eamle 9.09 (cotiued) The umber of equally likely samle oits i the samle sace is the umber of ways of drawig the radom samle of four crates from the twelve crates o the truck: 1 S C The umber of ways i which eactly crates of biders ca aear amog the four crates i the radom samle is (E) = (drawig biders from 3) (drawig the remaiig (4 ) crates from 9 aer) 3 9 C C4 The robability mass fuctio for X follows: P X 3 9 C C4 1 C4 The oly o-zero values of this.m.f. occur for = 0, 1, ad 3. It is imossible for all four crates i the samle to cotai biders, because there are oly three crates of biders i the etire oulatio ad the samle is take without relacemet C PX C C C3 1 1 C4 C4 [Note: if X had bee assumed to be biomial ( = 4, =.5), the PX.6 ]

15 ENGI 441 Discrete Probability Distributios Page 9-15 The geeral case for the hyergeometric.m.f. follows. If a oulatio of fiite size N cotais R successes ad (N R) failures, a radom samle of size is draw from this oulatio, ad X reresets the umber of successes i the samle, the PX R C N NR C C This robability mass is zero uless all of the followig are true: is a iteger i the rage [0, N] ad is a iteger i the rage [0, R] ad ( ) is a iteger i the rage [0, N R]. This distributio ca be obtaied from first riciles, usig coutig techiques, as i Eamle A Ecel sreadsheet for this distributio is available at " No-eamiable; for referece oly: The mea ad variace of a radom quatity that follows the hyergeometric distributio, X ~ H(, R, N) are R N EX ad I the limit as R ad N go to ifiity, such that R R N V X 1 N N N 1 R remais costat, the hyergeometric N ad 1. distributio becomes the biomial distributio b(, ), with As a geeral rule of thumb, the biomial distributio may be used as a accetably good aroimatio to the hyergeometric distributio if the samle size is o more tha 5% of the oulatio size N.

16 ENGI 441 Discrete Probability Distributios Page 9-16 Negative Biomial Probability Distributio This distributio will also be develoed by way of a eamle. Eamle 9.10: A motor is acceted if it starts at least twice i the first three attemts. O ay oe attemt, the robability of a success is.8, ideedetly of all other trials. Fid the robability that the motor is acceted. Let S = attemt is successful S ) F = attemt is usuccessful ( = E = the motor is acceted the E = S1S S1F S3 F1 SS 3 so P[E] = PS S PS F S PF S S [mutually eclusive evets] [ideedet evets] This is a eamle of the egative biomial distributio NB(; r, ), (Navidi, sectio 4.4, ages 34-37; Devore, sectio 3.5, ages ). I this case, with X = (umber of attemts eeded to obtai the secod success), X ~ NB(;,.8) ad P[E] = P[X 3] = P[X = ] + P[X = 3].

17 ENGI 441 Discrete Probability Distributios Page 9-17 The.m.f., mea ad variace of the egative biomial radom quatity X ~ NB(; r, ) are 1 r P 1,, 1,, r 1 r X r r r r E X, VX r 1 A secial case of the egative biomial distributio occurs whe r = 1: the geometric distributio, where X = the umber of trials u to ad icludig the first success ~ Geom(). 1 X P 1, 1 1 E X, VX Kowledge of the formulae o this age will ot be assumed i a test or the fial eamiatio. They are derived i the Navidi ad Devore tetbooks. Some ractice is rovided i the roblem sets.

18 ENGI 441 Discrete Probability Distributios Page 9-18 Poisso Distributio If the discrete radom quatity X is a cout of the umber of times a evet occurs i some fied iterval of time (or legth or area or volume), the the robability distributio for X may be Poisso. This situatio ca arise if the waitig time T from oe occurrece to the et is ideedet of all revious waitig times (see the eoetial distributio i Chater 10). The Poisso distributio ca also arise as a secial case of the biomial distributio, i the limit as the samle size icreases to ifiity while the eected umber of successes,, is held costat. Let, the the biomial.m.f. becomes 1 1! PX 1 1!!! ! ! lim P 11 1 lim 1 1 0! cost. X cost. From ENGI 345 (or equivalet), we kow that lim 1 Therefore lim b ;, lim C 1 cost. cost. The arameter is the mea umber of successes i the fied iterval. The Poisso robability mass fuctio is e e P Poisso ;, 0, 1,, 3,! e! X We shall see later that, for a Poisso radom quatity X,.

19 ENGI 441 Discrete Probability Distributios Page 9-19 Eamle 9.11 I a factory roducig isulated cables, the umber of cracks i the cables may be modelled to a good aroimatio as a Poisso rocess with a mea rate of occurrece of 3.4 cracks er metre. Fid the robability that (a) there is at least oe crack i a oe metre log cable. (b) there are less tha 0 cracks i a five metre log cable. (a) Let X = (umber of cracks i the oe-metre log cable) X ~ Poisso(3.4) X 1 PX 1 1 P 0 P 1 X ! e P X (b) Let Y = (umber of cracks i the five-metre log cable) If, o average, there are 3.4 cracks i every oe metre, the there are cracks o average i every five metres. Y ~ Poisso(17) 19 y 0 Y PY 19 P 0 y 17 e y! e ! This is a tedious calculatio by maual meas, but easy for a comuter to evaluate. We fid that P[Y < 0] = =.736 (3 s.f.).

20 ENGI 441 Discrete Probability Distributios Page 9-0 Eamle 9.11 (cotiued) The followig etracts from the Ecel sreadsheet file " ca be used istead. The value of the arameter of the Poisso distributio is 17 i art (b). P[Y < 0] = P[Y 19], so fid the value of the Poisso c.d.f. F (19; 17). 17.m.f. c.d.f. P[X = ] P[X ] If a questio ivolvig the calculatio of several adjacet Poisso robability masses arises i a test or eamiatio, the the aroriate table of values, similar to that show above, will be rovided with the questio aer.

21 ENGI 441 Discrete Probability Distributios Page 9-1 Mea ad Variace of a Poisso Radom Quatity This derivatio is ot eamiable, but the result is quotable. X e, the! If P e 1 (coherece coditio) ad! 0 1 E X e 0 e e!! 1! y Let y = 1, the X E e 1 y! y 0 1 E X e 0 e e!! 1! e e e 1! 1! 1! y 0 e e e!!! y y 0 Let t =, the t E X e t! t 0 V X E X E X Therefore

22 ENGI 441 Discrete Probability Distributios Page 9- Eamle 9.1 (Navidi tetbook, eercises 4.3, age 8, questio 10) A chemist wishes to estimate the cocetratio of articles i a certai susesio. She withdraws 3 ml of the susesio ad couts 48 articles. Estimate the cocetratio i articles er ml ad fid the ucertaity i the estimate. We assume that the cocetratio c (umber of articles er ml) is costat across all volumes. The the umber N of articles i volume v has the distributio N ~ Poisso, where cv. We do ot kow the true cocetratio c with certaity. We do have a oit estimate, 48 cˆ 16 ml 1 v 3mL N ~ Poisso cv The arameter of the Poisso distributio is the estimated by ˆ cv ˆ But for ay radom quatity N that follows a Poisso distributio, VN EN N VN 48 VC V V N v v v 9 3 The ucertaity i the estimate of cocetratio is c, which is ukow, but ca be estimated 16 4 from the data as c c v 3 Therefore the estimated cocetratio is c articles / ml to 1 d.. I geeral, whe articles are couted i a volume v, the estimate of cocetratio is c v v articles / ml cˆ cˆ v For a fied cocetratio c, quadrulig the volume halves the ucertaity i c.

23 ENGI 441 Discrete Probability Distributios Page 9-3 Eamle 9.13 Shis arrive at a ort o average every 30 miutes, ideedetly of all revious arrivals. No two shis ever arrive at eactly the same time. Fid the robability that, durig the et hour, (a) o shis arrive; (b) more tha four shis arrive. Shis arrive, o average, at a rate 1shi mi mi hr shis er hour. Therefore the umber of shis N that will arrive i the et hour is a radom quatity that follows the distributio N ~ Poisso(). 0 P N e e ! (a) 0 (b) PN 4 1 PN 4 1 Poisso 4; From the Ecel sreadsheet file, Poisso(4; ) = or 1 P N 4 1 e e 1 e P N [Ed of Chater 9]

24 ENGI 441 Discrete Probability Distributios Page 9-4 [Sace for Additioal Notes]