Problm Statmnt Suppos small arriv at a crtain airport according to Poisson procss with rat α pr hour, so that th numbr of arrivals during a tim priod of t hours is a Poisson rv with paramtr t (a) What is th probability that actly 5 small arriv during a 1-hour priod? At last 5? At last 1? (b) What ar th pctd valu and standard dviation off th numbr of small that arriv during a 9-min priod? (c) What is th probability that at last 2 small arriv during a 2 ½ hour priod? That at most 1 arriv during this priod? Dfinitions, Equations and Hlpful Hints Poisson Probability Distribution is dfind by th random variabl X(t) A numbr of succsss in a continuum (tim or spac). Th rv X(t) is said to b a Poisson distribution or X(t) ~ Poisson[ αt] (pags 12-131 in tt). Th paramtr is dfind: Th pmf of X(t) is dfind: Th cdf of X(t) is dfind: Th pctd valu of X(t) is dfind: Th varianc of X(t) is dfind: αt p[ ; ] F[ ; ]! E[ X ( t)] αt! V[ X ] αt, 1, 2, K o/w Th standard dviation is dfind: σ [ X ( t)] V[ X ( t)] αt Nots: If w hav any binomial primnt in which n is larg and p is small b[;n,p] p[;] whr np. This approimation can b safly applid if n 1, p.1, and np 2 (pag 129 in tt). Valus for th cumulativ distribution, F[X(t)], of th Poisson can b found in Tabl 2A on pag 73 in th tt. 1 OF 5
Solutions Dfin: Givn: X Numbr of that arriv in t hours a /hour X ~ poisson[l at t] (a) What is th probability that actly 5 small arriv during a 1-hour priod? At last 5? At last 1? (*mans s not) This problm has thr parts. Each part involvs using th Poisson distribution pmf dfinition howvr, varis in ach part. Th first part of th qustion is solvd using th Poisson distribution pmf dfinition, whr t 1 hour and 5. αt ( 1hour) t 5 () X( t 1) 5].92 t! 5! Nt, th problm rmains th sam cpt 5. Th complmnt is usd to find th solution: αt ( 1hour) 4 X ( t 1) 5] 1 F(4) 1! 1 * From Tabl 2A pag 73 in tt. 4! 1.1*.9 Again th problm is similar cpt 1. Th complmnt is also usd to find th solution: αt ( 1hour) 9 X ( t 1) 1] 1 F(9) 1! 1 9 * From Tabl 2A pag 73 in tt.! 1.717*.23 2 OF 5
(b) What ar th pctd valu and standard dviation of th numbr of small that arriv during a 9-min priod? Not that 9min 1.5hours. Th pctd valu of th Poisson distribution is calculatd from th dfinition mntiond in Dfinitions, Equations and Hlpful Hints sction: E[ X( t 1.5] αt ( 1.5 hours) 12 Th standard dviation of this Poisson distribution is calculatd by: σ[ X ] V[ ] 12 2 3.4641 (c) What is th probability that at last 2 small arriv during a 2 ½ hour priod? That at most 1 arriv during this priod? This problm has two parts. Each part utilizs th Poisson distribution pmf dfinition howvr th valu of varis. In th first part t 2.5 hours and 2. Th complmnt is usd to find th solution. αt X ( t 2.5) 2] 1 F(19) * From Tabl 2A pag 73 in tt. ( 2.5 hours) 2 19 1 1! 1.47*.53 19 2 2! Now 1 and Poisson distribution pmf dfinition is usd to find th solution. X ( t 2.5) 1] F(1) 1! 1 2 2!.11* * From Tabl 2A pag 73 in tt. 3 OF 5
Using Microsoft EXCEL Microsoft s EXCEL spradsht contains intrinsic functions to solv Binomial and Poisson distribution pmf problms. To us this proprty of EXCEL, click on to a cll in th spradsht. This cll will contain th rsult. Th intrinsic functions ar containd in th function wizard mnu. This mnu can b accssd by clicking on th f() button on th toolbar or by pulling down th Insrt mnu and scrolling down to Function. A function window should appar. For Binomial distribution applications Slct Statistical in th function catgory and BINODIST in th function nam catgory. Click OK. A window should appar asking for th following information: o Numbr_s th numbr of succssful trials, o Trials th numbr of trials, n o Probability_s th probability of a succss, p() o Cumulativ tru or fals. Tru if th cumulativ distribution function is to b calculatd. Fals for th convrs. For Poisson distribution applications Slct Statistical in th function catgory and POISSON in th function nam catgory. Click OK. A window should appar asking for th following information: o X th numbr of vnts, n o Man th pctd valu of, X o Cumulativ tru or fals. Tru if th cumulativ distribution function is to b calculatd. Fals for th convrs. Following ths stps and corrctly answring th function rquirmnts should giv a rsult. For ampl, Figur 1 is a graph comparing diffrnt Poisson distribution pmfs with varying valus of. Th pmf valus in th following graph ar th rsults from th EXCEL intrinsic function POSSION. Th graph s trnd shows: as incrass th maimal probability occurs with an incrasd numbr of succsss in a tim priod. Th pmfs also bcom incrasingly mor distributd as incrass. 4 OF 5
.16.14.12 lambda (t1) lambda12 (t1.5) lambda2 (t2.5).1 X(t)]..6.4.2 5 1 15 2 25 3 35 No. of that arriv in t hours Figur 1. Comparison of th Poisson distribution pmf with varying valus of. 5 OF 5