4. The Poisson Distribution


 Brice Farmer
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1 Virual Laboraories > 13. The Poisson Process > The Poisson Disribuion The Probabiliy Densiy Funcion We have shown ha he k h arrival ime in he Poisson process has he gamma probabiliy densiy funcion wih shape parameer k and rae parameer r: f *k ( ) = r k k 1 (k 1)! e r, 0 Recall also ha a leas k arrivals come in he inerval ( 0, ] if and only if he k h arrival occurs by ime : 1. Use inegraion by pars o show ha ( N k) (T k ) P( N k) = f * k 0 ( s ) ds = 1 k 1 e r ( r ) j, k N j =0 j! 2. Use he resul of Exercise 1 o show ha he probabiliy densiy funcion of he number of arrivals in he inerval ( 0, ] is P( N = k) = e r ( r ) k k!, k N The corresponding disribuion is called he Poisson disribuion wih parameer r ; he disribuion is named afer Simeon Poisson. 3. In he Poisson experimen, vary r and wih he scroll bars and noe he shape of he densiy funcion. Now wih r = 2 and = 3, run he experimen 1000 imes wih an updae frequency of 10 and wach he apparen convergence of he relaive frequency funcion o he densiy funcion. The Poisson disribuion is one of he mos imporan in probabiliy. In general, a discree random variable N in an experimen is said o have he Poisson disribuion wih parameer c > 0 if i has he probabiliy densiy funcion g(k) = e c c k 4. Show direcly ha g is a valid probabiliy densiy funcion. 5. Show ha k!, k N g(n 1) < g(n) if and only if n < g a firs increases and hen decreases, and hus he disribuion is unimodal If c is no an ineger, here is a single mode a c. If c is an ineger here are wo modes a c 1 and
2 6. Suppose ha requess o a web server follow he Poisson model wih rae r = 5. per minue. Find he probabiliy ha here will be a leas 8 requess in a 2 minue period. 7. Defecs in a cerain ype of wire follow he Poisson model wih rae 1.5 per meer. Find he probabiliy ha here will be no more han 4 defecs in a 2 meer piece of he wire. Momens Suppose ha N has he Poisson disribuion wih parameer The following exercises give he mean, variance, and probabiliy generaing funcion of N. 8. Show ha E( N ) = 9. Show ha var( N ) = 10. Show ha E(u N ) = e c (u 1). for u R. Reurning o he Poisson process {N : 0} wih rae parameer r, i follows ha E( N ) = r and var( N ) = r for 0. Once again, we see ha r can be inerpreed as he average arrival rae. In an inerval of lengh, we expec abou r arrivals. 11. In he Poisson experimen, vary r and wih he scroll bars and noe he locaion and size of he mean/sandard deviaion bar. Now wih r = 3 and = 4, run he experimen 1000 imes wih an updae frequency of 10 and wach he apparen convergence of he sample mean and sandard deviaion o he disribuion mean and sandard deviaion, respecively. 12. Suppose ha cusomers arrive a a service saion according o he Poisson model, a a rae of r = 4. Find he mean and sandard deviaion of he number of cusomers in an 8 hour period. Saionary, Independen Incremens Le us see wha he basic regeneraive assumpion of he Poisson process means in erms of he couning variables {N : 0}. 13. Show ha if s <, hen N N s is he number of arrivals in he inerval ( s, ]. Recall ha our basic assumpion is ha he process essenially sars over a ime s and he behavior afer ime s is independen of he behavior before ime s. 14. Argue ha: N N s has he same disribuion as N s namely Poisson wih parameer r ( s). N N s and N s are independen.
3 15. Suppose ha N and M are independen random variables, and ha N has he Poisson disribuion wih parameer c and M has he Poisson disribuion wih parameer d. Show ha N + M has he Poisson disribuion wih parameer c + d. Give a probabilisic proof, based on he Poisson process. Give an analyic proof using probabiliy densiy funcions. Give an analyic proof using probabiliy generaing funcions. 16. In he Poisson experimen, selec r = 1 and = 3. Run he experimen 1000 imes, updaing afer each run. By compuing he appropriae relaive frequency funcions, invesigae empirically he independence of he random variables N 1 and N 3 N 1. Normal Approximaion Now noe ha for k N +, N k = N 1 + ( N 2 N 1 ) + + ( N k N k 1 ) The random variables in he sum on he righ are independen and each has he Poisson disribuion wih parameer r. 17. Use he cenral limi heorem o show ha he disribuion of he sandardized variable below converges o he sandard normal disribuion as k. Z k = N k k r A bi more generally, he same resul is rue wih he ineger k replaced by he posiive real number Thus, if N has he Poisson disribuion wih parameer c, and c is large, hen he disribuion of N is approximaely normal wih mean c and sandard deviaion since he Poisson is a discree disribuion. k r When using he normal approximaion, we should remember o use he coninuiy correcion, 18. In he Poisson experimen, se r = 1 and = 1. Increase r and and noe how he graph of he probabiliy densiy funcion becomes more bellshaped. 19. In he Poisson experimen, se r = 5 and = 4. Run he experimen 1000 imes wih an updae frequency of 100. Compue and compare he following: P(15 N 4 22) The relaive frequency of he even {15 N 4 22}. The normal approximaion o P(15 N 4 22). 20. Suppose ha requess o a web server follow he Poisson model wih rae r = 5. Compue he normal approximaion o he probabiliy ha here will be a leas 280 requess in a 1 hour period.
4 Condiional Disribuions Consider again he Poisson model wih arrival ime sequence (T 1, T 2,...) and couning process {N : 0}. 21. Le > 0. Show ha he condiional disribuion of T 1 given N = 1 is uniform on he inerval ( 0, ). Inerpre he resul. 22. More generally, show ha given N = n, he condiional disribuion of (T 1,..., T n ) is he same as he disribuion of he order saisics of a random sample of size n from he uniform disribuion on he inerval ( 0, ). Noe ha he condiional disribuion in he las exercise is independen of he rae r. This resul means ha, in a sense, he Poisson model gives he mos random disribuion of poins in ime. 23. Suppose ha requess o a web server follow he Poisson model, and ha 1 reques comes in a five minue period. Find he probabiliy ha he reques came during he firs 3 minues of he period. 24. In he Poisson experimen, se r = 1 and = 2. Run he experimen 1000 imes, updaing afer each run. Compue he appropriae relaive frequency funcions and invesigae empirically he heoreical resul in Exercise Suppose ha 0 < s < and ha n is a posiive ineger. Show ha he condiional disribuion of N s given N = n is binomial wih rial parameer n and success parameer p = s. Noe ha he condiional disribuion is independen of he rae r. Inerpre he resul. 26. Suppose ha requess o a web server follow he Poisson model, and ha 10 requess come during a 5 minue period. Find he probabiliy ha a leas 4 requess came during he firs 3 minues of he period. Esimaing he Rae In many pracical siuaions, he rae r of he process in unknown and mus be esimaed based on observing he number of arrivals in an inerval. 27. Show ha E N = r and hence N is an unbiased esimaor of r. Since he esimaor is unbiased, he variance measures he mean square error of he esimaor. 28. Show ha var N = r and hence var N 0 as. This means ha N is an consisen esimaor of r. 29. In he Poisson experimen, se r = 3 and = 5. Run he experimen 100 imes, updaing afer each run. For each run, compue he esimae of r based on N. Over he 100 runs, compue he average of he squares of he errors. Compare he resul in (b) wih he variance in Exercise Suppose ha requess o a web server follow he Poisson model wih unknown rae r per minue. In a one hour
5 period, he server receives 342 requess. Esimae r. Virual Laboraories > 13. The Poisson Process > Conens Apples Daa Ses Biographies Exernal Resources Keywords Feedback
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