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# 389 Class Note for STAT 416 at PSU

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This 19 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Pennsylvania State University taught by a professor in Fall. Since its upload, it has received 27 views.

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Date Created: 02/06/15

Exponential Distribution 0 De nition Exponential distribution with parameter A Ae x20 fltx0 xlt0 oThecdf 75 1 6 5620 FmfMw0 xlt0 0 Mean EX1 o Moment generating function EW ngtltA EX2 g gq5tt0 2 w moor EX2 EX2 12 0 Properties 1 Memoryless PX gt s t X gt t PX gt s PX gt sth gt t PX gt stX gt t PX gt15 PX gt s t PX gt15 e Ms f e At e As PX gt s Example Suppose that the amount of time one spends in a bank is exponentially distributed with mean 10 minutes llO What is the prob ability that a customer will spend more than 15 minutes in the bank What is the probability that a customer will spend more than 15 min utes in the bank given that he is still in the bank after 10 minutes Solution PX gt 15 e W 6 32 022 PX gt15lX gt10 PX gt 5 6 12 0604 Failure rate hazard rate function Ht 7 f t W i 1 Ft PX E tt dtX gt t rtdt For exponential distribution Mt A t gt 0 Failure rate function uniquely determines Ft Ft 1 6 f5 W 2 If X z 12 n are iid exponential RVs With mean l the pdf of 2211 X is Atyl l leX2 Xnt A6 Alton 7 gamma distribution with parameters n and A 3 If X1 and X2 are independent exponential RVs with mean l1 l2 1 1 2 4 If X 2 12 n are independent exponential RVs with rate m Let Z minX1 Xn and Y maxltX1 X Find distribution of Z and Y Z is an exponential RV with rate 2211 m PZ gt 51 PminX1 Xn gt L gt X2 gt L Xn gt 1 PX1 gt 51PX2 gt x PXn gt x n H e Mfr e ZQQMW Z39l PX1 lt X2 was PltY lt as M1 Poisson Process 0 Counting process Stochastic process Ntt Z 0 is a counting process if N t represents the total num ber of events that have occurred up to time t N t 2 0 and are of integer values N t is nondecreasing in t 0 Independent increments the numbers of events oc curred in disjoint time intervals are independent 0 Stationary increments the distribution of the number of events occurred in a time interval only depends on the length of the interval and does not depend on the position o A counting process N tt 2 0 is a Poisson pro cess with rate A A gt 0 if 1 N 0 0 2 The process has independent increments 3 The process has staionary increments and N ts N 3 follows a Poisson distribution with parameter At W PNts Ns n 6 n n 0 1 0 Note ENt 8 Ns At ENt ENt 0 N0 At Interarrival and Waiting Time 0 De ne Tn as the elapsed time between 71 lst and the nth event Tm n l 2 is a sequence of inierarrival times 0 Proposition 51 Tn n 12 are independent identically distributed exponential random variables with mean l 0 De ne S as the waiting time for the nth event ie the arrival time of the nth event 11 0 Distribution of Sn At MW 1 71 l gamma distribution with parameters n and A 0 ESn ZZZ1 ETz39 nA 9105 0 Example Suppose that people immigrate into a terri tory at a Poisson rate l per day a What is the expected time until the tenth immigrant arrives b What is the probability that the elapsed time between the tenth and the eleventh arrival exceeds 2 days Solution Time until the 10th immigrant arrives is Sm ESio lO 10 PT11 gt 2 e 2A 0133 Further Properties 0 Consider a Poisson process N t t 2 0 with rate A Each event belongs to two types I and II The type of an event is independent of everything else The probability of being in type I is p 0 Examples female vs male customers good emails vs spams 0 Let N1t be the number of type 1 events up to time t 0 Let N2t be the number of type 11 events up to time t N05 N1t N205 0 Proposition 52 N1tt 2 0 and N2tt 2 0 are both Poisson processes having respective rates A and 1 p Furthermore the two processes are in dependent Example If immigrants to area A arrive at a Poisson rate of 10 per week and if each immigrant is of En glish descent with probability 1 12 then what is the probability that no people of English descent will im migrate to area A during the month of February Solution The number of English descent immigrants arrived up to time t is N1t which is a Poisson process with mean 12 1012 PltN1lt4gt o eWW 10 0 Conversely Suppose N1tt 2 0 and N2tt 2 0 are independent Poisson processes having respec tive rates 1 and A2 Then Nt N1t N2t is a Poisson process with rate 1 2 For any event occurred with unknown type independent of every thing else the probability of being type I is p A1132 and type II is 1 p 0 Example On a road cars pass according to a Poisson process with rate 5 per minute Trucks pass accord ing to a Poisson process with rate 1 per minute The two processes are indepdendent If in 3 minutes 10 veicles passed by What is the probability that 2 of them are trucks Solution Each veicle is independently a car with probability g and a truck with probability The probabil ity that 2 out of 10 veicles are trucks is given by the binomial distribution 12 28 ll Conditional Distribution of Arrival Times 0 Consider a Poisson process N tt 2 0 with rate A Up to t there is exactly one event occurred What is the conditional distribution of T1 0 Under the condition T1 uniformly distributes on 0 t oPl OOf PT1 lt Suva 1 i PT1 lt sNt 1 7 POW PNs 1 N05 Ns 0 PNt 1 PNs PNt Ns0 PN t 1 Ase As e Mt s Ate At Note cdf of a uniform 12 o If N t n what is the joint conditional distribution of the arrival times 31 SQ Sn 0 1 SQ Sn is the ordered statistics of n independent random variables uniformly distributed on 0 t 0 Let Y1 Y2 Yn be n RVs Ya 32 Ym is the ordered statistics of Y1 Y2 Yn if lk is the kth smallest value among them 0 If Yi 2 l n are iid continuous RVs with pdf f then the joint density of the ordered statistics Y Ye Ym lS fi17Y277Yny17 327 quot7 311 quotlH1fy 21 lt 22 lt lt Km 0 otherwise 13 0 We can show that n flt817827 78n Proof f3132 sn l Nt 71 sh 32 3n n POW 71 Ae AslAe MSQ sl Ae Asn sn1 t sn e Mt nl n o For n independent uniformly distributed RVs on 0 t Y1 Y 1 fy17 327 quot397 t n 0 Proposition 54 Given Sn t the arrival times 31 SQ SW1 has the distribution of the ordered statis tics of a set n 1 independent uniform 0 t random variables 14 Generalization of Poisson Process 0 Nonhomogeneous Poisson process The counting pro cess N tt 2 0 is said to be a nonhomogeneous Poisson process with intensity function t t 2 0 if 1 N 0 0 2 The process has independent increments 3 The distribution of N t s N t is Poisson with mean given by mt s mt where 0 We call mt mean value function 0 Poisson process is a special case where t A a constant 15 0 Compound Poisson process A stochastic process X tt 2 0 is said to be a compound Poisson pro cess if it can be represented as il where N tt 2 0 is a Poisson process and Ym 2 0 is a family of independent and identically distributed random variables which are also indepen dent ofNtt 2 0 o The random variable X t is said to be a compound Poisson random variable 0 Example Suppose customers leave a supermarket in accordance with a Poisson process If E the amount spent by the 2th customer 239 12 are indepen dent and identically distributed then X t Egg Yi the total amount of money spent by customers by time t is a compound Poisson process 16 c Find E X 75 and VarX o EXt MEG1 o VarXt tVa7 Y1 a Proof PNt nEXtNt ZPNt nnEY1 Egg 1 nPNt n EY EENOL D 17 VarXt Nt n 18 VarXt 13X2lttgtgt ltEltXlttgtgtgt2 Z PNt nEX2tWt n EXt2 Z PNt nnVarY1 n2E2Y1 MEG12 VarY1ENt E2ltY1EN2ltt MEG12 AtVarOl tE2Y1 tVa7 Y1 MEGf 19

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