Lecture 9. Some Useful Discrete Distributions. Some Useful Discrete Distributions. The observations generated by different experiments have

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1 NM 7 Lecture 9 Some Useful Dscrete Dstrbutos Some Useful Dscrete Dstrbutos The observatos geerated by dfferet eermets have the same geeral tye of behavor. Cosequetly, radom varables assocated wth these eermets ca be descrbed by the same robablty dstrbuto ad therefore ca be rereseted by a sgle formula

2 Dscrete Uform Dstrbuto The smlest of all dscrete robablty dstrbutos s oe where the radom varable assumes each of ts values wth a equal robablty. Such a robablty dstrbuto s called as a dscrete uform dstrbuto. Defto : If the radom varable assumes the values,,, wth equal robabltes, the the dscrete uform dstrbuto s gve by, o.w.,..., Dscrete Uform Dstrbuto amle Whe a de s tossed, each elemet of the sace S {,, 3, 4, 5, 6} occurs wth robablty /6. Therefore, we have a uform dstrbuto, wth

3 Dscrete Uform Dstrbuto amle Whe a lght bulb s selected at radom from a bo that cotas a 4-watt bulb, a 6-watt bulb, a 75-watt bulb, a -watt bulb, each elemet of the samle sace S {4, 6, 75, } occurs wth robablty ¼. Therefore, we have a uform dscrete dstrbuto, wth Dscrete Uform Dstrbuto The mea of the dscrete uform dstrbuto s;

4 The varace of the dscrete uform dstrbuto s; [ ] σ σ σ Dscrete Uform Dstrbuto amle For the eermet of tossg a de, fd mea ad varace of the radom varable Dscrete Uform Dstrbuto

5 Beroull Dstrbuto There are may roblems whch the eermet cossts of trals or subeermets. Here, we are cocered wth a dvdual tral that has two ossble outcomes success {A} or falure {Ā}. Defto : A beroull tral results oe of two outcomes deoted A for success ad Ā for falure. For eamle, tossg a co s a Beroull tral, sce oly oe of two dfferet outcomes ca occur head H ad tal T. Beroull Dstrbuto Defto : A beroull radom varable,, s defed as the umercal outcome of a Beroull tral where f a success occurs ad f a falure occurs Cosequetly the robablty dstrbuto for s gve as follows, o.w. : robablty of success for a Beroull tral

6 Beroull Dstrbuto The mea of Beroull radom varable s; + + Beroull Dstrbuto The varace of Beroull radom varable s; [ ] + σ σ σ

7 Bomal Dstrbuto Cosder a eermet cossts of deedet beroull trals. There are oly two ossble outcomes o each tral, A for success ad Ā for falure. Suose that the robablty of a success s costat from tral to tral ad A, Ā -. Let the radom varable be defed as follows : umber of successes deedet beroull trals. sa bomal radom varable wth arameters ad. Bomal Dstrbuto Defto : A Beroull tral ca result a success wth robablty ad a falure wth robablty q-. The the robablty dstrbuto of the bomal radom varable,, the umber of successes deedet trals, s q,,,..., o.w.

8 Bomal Dstrbuto The arameters of the bomal dstrbuto are ad. A smle dervato of the bomal dstrbuto s: Let P{ successes trals} The robablty of the artcular outcome S wth successes for the frst trals ad falures for the last - trals s q AAA AAAA AAA P There are outcomes havg eactly successes ad - falures. Therefore!!! q Bomal Dstrbuto [ ] + q q P The Bomal dstrbuto s a robablty dstrbuto sce

9 Bomal Dstrbuto If we wat to ow the robabltes that fls of a balaced co wll yeld 5 heads ad 7 tals, 3 of ersos wll resod to a gve mal questoare, of 5 tems wll be defectve, there are 6 boys ad 9 grls amog 5 chldre bor o a gve day, I each case, we are terested the robablty of gettg a certa umber of successes a gve umber of trals. Bomal Dstrbuto The mea of Bomal radom varable s;!!!!!! [ + q] q q

10 Bomal Dstrbuto The varace of Bomal radom varable s; [ ] q q q q ] [!!!!!! σ σ Aother aroach to fd the mea ad varace s to cosder as a sum of deedet radom varables, each wth mea ad varace q, so that Var q + q + + q q Bomal Dstrbuto

11 Bomal Dstrbuto amle Suose that a rado tube serted to a certa tye of set has a robablty of. of fuctog more tha 5 hours. If we test tubes, what s the robablty that eactly of them fuctog more tha 5 hours,,,,, Bomal Dstrbuto amle The Natoal Foudato reorts that 7% of the U.S. graduate studets who ear PhD degrees egeerg are foreg atoals. Cosder the umber of foreg studets Y a radom samle of 5 egeerg studets who recetly eared ther PhD a Fd PY b Fd PY 5 c Fd ad σ.

12 The Geometrc Dstrbuto The geometrc dstrbuto s also based o a sequece of Beroull trals ecet that the umber of trals s ot fed, The radom varable of terest, deoted by, s defed to be the umber of trals requred to reach the frst success. The samle sace ad rage sace for are llustrated fgure followed. The Geometrc Dstrbuto The rage sace for s R{,,3,...} ad the dstrbuto of gve by q,, 3,.. ow.. s the umber of trals utl the frst success s observed.

13 The Geometrc Dstrbuto It s easy to verfy that ths s a robablty dstrbuto sce, a b for all q P q q q The mea ad varace of geometrc dstrbuto are resectvely P q d d q q dq dq q q The Geometrc Dstrbuto σ + q q q q + q q q + d + dq q q q d + q q q dq dq d q d q dq q dq q q + d q + dq q q q + q + σ d d dq q + q + q q

14 The Geometrc Dstrbuto amle A maufacturer uses electrcal fuses a electroc system. The fuses are urchased large lots ad tested sequetally utl the frst defectve fuse observed. Assume that the lot cotas % defectve fuses. What s the robablty that the frst defectve fuse wll be oe of the frst fve fuses tested? Fd the mea, varace ad stadard devato of, the umber of fuses tested utl the frst defectve fuse s observed. Negatve Bomal Dstrbuto The egatve bomal dstrbuto s also based o the Beroull trals. It s a logcal eteso of the geometrc dstrbutos. Radom varable deotes the umber of trals utl rth success s observed r q r r r, r +, r +,... o. w.

15 Negatve Bomal Dstrbuto r r q The term arses from the robablty assocated wth eactly oe outcome samle sace that has -r falures r success. I order for ths outcome to occur, there must be r- successes the - reettos before the last outcome, whch s always success. r There are arragemets satsfyg ths codto. Negatve Bomal Dstrbuto The mea of egatve bomal dstrbuto; r The varace of egatve bomal σ Var rq

16 Negatve Bomal Dstrbuto amle I a NBA chamosh seres, the team whch ws four games out of seve wll be the wer. Suose that the team A has robablty.55 of wg over the team B ad both team A ad B face each other the chamosh games. a What s the robablty that team A wll w the seres s games? b What s the robablty that team A wll w the seres? c If both teams face each other a regoal layoff seres ad the wer s decded by wg three out of fve games, what s the robablty that team A wll w a layoff? Multomal Dstrbuto The bomal eermet becomes multomal eermet f each tral has more tha outcomes. The classfcato of a maufactured roducts as beg lght, heavy, or accetable The recordg of accdets at a certa tersecto accordg to the day of wee costtute multomal eermet. If a gve tral ca result ay oe of ossble outcomes,,, wth robabltes,,,, the the multomal dstrbuto gve the robablty that occurs tmes, occurs tmes,, occurs tmes deedet trals, where , sce the results of each tral must be oe of the ossble outcomes.

17 Multomal Dstrbuto Sce the trals are deedet, ay secfed order yeldg outcomes for, for,, for wll occur wth robablty,,,. The total umber of orders yeldg smlar outcomes for the trals s equal to the umber of arttos of tems to grous wth the frst grou; the secod grou,, the th grou. Ths ca be doe!,,...,!!...! Sce all the arttos are mutually eclusve ad occur wth equal robablty, the multomal dstrbuto s obtaed by multlyg the robablty for a secfed order by the total umber of arttos. Multomal Dstrbuto!,,...!!...!...,,,...; o. w.,,,...;...;,,,... where The mea ad varace of are Var

18 Multomal Dstrbuto amle For a arort cotag three ruways t s ow that the deal settg the followg are the robabltes that the dvdual ruways are accessed by radom arrvg commercal jet Ruway : /9 Ruway : /6 Ruway 3: 3 /8 What s the robablty that 6 radomly arrvg arlaes are dstrbuted followg fasho? Ruway : arlaes Ruway : arlaes Ruway 3: 3 arlaes The Hyergeometrc Dstrbuto The dstcto betwee the bomal dstrbuto ad hyergeometrc dstrbuto les the way the samlg s doe. The tyes of alcatos of the hyergeometrc are very smlar to the bomal dstrbuto. We are terested comutg robabltes for the umber of observatos that fall to a artcular category. But, the case of the bomal, deedece amog trals are requred. If the bomal s aled to samlg from a lot of tems dec of cards, batch of roducto tems, the samlg must be doe wth relacemet of each tem after t s observed. The hyergeometrc dstrbuto does ot requre deedece ad s based o the samlg doe wthout relacemet.

19 The Hyergeometrc Dstrbuto The hyergeometrc eermet has two roertes: A radom samle of sze s selected wthout relacemet from N tems of the N tems may be classfed as successes ad N- are classfed as falures. The radom varable,, s the umber of successes o a hyergeometrc eermet...,,,..., w o N N The Hyergeometrc Dstrbuto The mea ad varace of hyergeometrc dstrbuto are N N N Var N.. σ

20 The Hyergeometrc Dstrbuto amle: Lots of 4 comoets each are called uaccetable f they cota as may as 3 defectves or more. The rocedure for samlg the lot s to select 5 comoets at radom ad to reject the lot f a defectve s foud. What s the robablty that eactly defectve s foud the samle f there are 3 defectve the etre lot? Fd mea ad varace of defectve comoets. The Hyergeometrc Dstrbuto amle: From a lot of mssles, 4 are selected at radom ad fred. If lot cotas 3 defectves mssles that wll ot fre, what s the robablty that a All 4 wll fre? b At most wll ot fre?

21 Refereces Walole, Myers, Myers, Ye,, Probablty & Statstcs for geers & Scetsts Degz, B., 4, Lecture Notes o Probablty, htt://w3.gaz.edu.tr/web/bdegz Hes, Motgomery, 99, Probablty & Statstcs geerg & Maagemet Scece