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697584250 697584250 Presentation Transcript

  • Outline Exponential distribution Counting Process Poisson Process Nonhomogeneous PP Compound PP AMA528 PROBABILITY AND STOCHASTIC MODELS DEPARTMENT OF APPLIED MATHEMATICS Lecturer & Tutor: Dr. Catherine LIU Contact: 2766 6931 (O); Office Venue: HJ616 Consultation Hours: 7:45pm-8:45pm, Mon. & 4:00pm-5:00pm, Tues. 16/11/2011 AMA528 (By Catherine Liu) Lecture 10 The Poisson Process 16/11/2011 1 / 15
  • Outline Exponential distribution Counting Process Poisson Process Nonhomogeneous PP Compound PPChapter 7The Poisson Process AMA528 (By Catherine Liu) Lecture 10 The Poisson Process 16/11/2011 2 / 15
  • Outline Exponential distribution Counting Process Poisson Process Nonhomogeneous PP Compound PPOutline 1 Review of exponential Distribution 2 Counting Process 3 Poisson Process 4 Nonhomogeneous Poisson Process 5 Compound Poisson Process AMA528 (By Catherine Liu) Lecture 10 The Poisson Process 16/11/2011 3 / 15
  • Outline Exponential distribution Counting Process Poisson Process Nonhomogeneous PP Compound PPReview on exponential distribution pdf: 0-rate parameter, Y exp ; 0. y 1 x e if y 0 e if x 0 f y g x 0 elsewhere 0 elsewhere 2 2 E Y 1 Var Y 1 ;E X Var X . cdf: y x 1 e if y 0 1 e if x 0 F y G x 0 elsewhere 0 elsewhere t 1 1 MGF: MY t 1 ; MX s 1 s . Memoryless: For all s t 0, P X s tX t P X s or P X s t P X s P X t . AMA528 (By Catherine Liu) Lecture 10 The Poisson Process 16/11/2011 4 / 15
  • Outline Exponential distribution Counting Process Poisson Process Nonhomogeneous PP Compound PPA noteSuppose X1 Xn i.i. and Xi exp i for i 1 n. n P min X1 Xn x P X1 x exp i x i 1 i P Xi Xj i jEg1 (example 5.5, pp.2): Suppose one has a stereo system consisting of two mainparts, a radio and a speaker. If the lifetime of the radio is exponential with mean 1000hours and the lifetime of the speaker is exponential with mean 500 hours independentof the radio’s lifetime, then what is the probability that the system’s failure (when itoccurs) will be caused by the radio failing? AMA528 (By Catherine Liu) Lecture 10 The Poisson Process 16/11/2011 5 / 15
  • Outline Exponential distribution Counting Process Poisson Process Nonhomogeneous PP Compound PPCounting ProcessA stochastic process N t t 0 is said to be a counting process if N t representsthe total number of ”events” that have occurred up to time t if 1 N t 0; 2 N t is integer valued; 3 If s t, then N s N t ; 4 For s t, N t N s equals the number of events that have occurred in the interval s t .Eg: Let N t equal # of persons who enter 7-11 shop at or prior to (or by) time t, then N t t 0 is a counting process;But if N t equal # of persons in the store at time t, then N t t 0 would not be acounting process.Independent increments: if # of events that occur in disjoint time intervals areindependent.Eg: N 10 N 3 is independent of N 15 N 10 . AMA528 (By Catherine Liu) Lecture 10 The Poisson Process 16/11/2011 6 / 15
  • Outline Exponential distribution Counting Process Poisson Process Nonhomogeneous PP Compound PPPoisson ProcessA counting process N t t 0 is said to be a Poisson Process having rate 0,if 1 N 0 0; 2 The process has independent increments; 3 The # of events in any interval of length t is Poisson distributed with mean t. That is, for all s t 0, n t t P N t s N s n e n 0 1 n AMA528 (By Catherine Liu) Lecture 10 The Poisson Process 16/11/2011 7 / 15
  • Outline Exponential distribution Counting Process Poisson Process Nonhomogeneous PP Compound PPAlternative definition f x f x Infinite small o 1 : f x o x or x o 1 or lim x 0. x 0A counting process N t t 0 is said to be a Poisson Process having rate 0,if 1 N 0 0; 2 The process has independent and stationary increments; 3 P N h 1 h o h ; 4 P N h 2 o h . AMA528 (By Catherine Liu) Lecture 10 The Poisson Process 16/11/2011 8 / 15
  • Outline Exponential distribution Counting Process Poisson Process Nonhomogeneous PP Compound PPInterarrival and waiting timesConsider a Poisson process: Let T1 be the time of the first event; Let Tn denote theelapsed time between the n 1 st and the nth event for n 1.The sequence Tn n 1 2 is called the sequence of interarrival times.Distribution of Tn : Tn n 1 2 i.i.d. exp . nLet Sn Ti , n 1, the arrival time of the n-th event, then Sn is called i 1the waiting time until the n-th event.Distribution of Sn : Sn Gammar n . The pdf of Sn is n 1 t t fsn t e I t 0 n 1A useful result: Sn t N t nRemark: Based on Tn n 1 with rate , we can set up a Poisson process with rate . AMA528 (By Catherine Liu) Lecture 10 The Poisson Process 16/11/2011 9 / 15
  • Outline Exponential distribution Counting Process Poisson Process Nonhomogeneous PP Compound PPProperty 1Suppose two Poisson processes 1 N1 t t 0 with rate 1 ; Sn : the time of the n-th event of the 1st process; 2 N2 t t 0 with rate 2; Sm : the time of the m-th event of the 2nd process. N1 t t 0 and N2 t t 0 are independent. n m 1 k n m 1 k 1 2 n m 1 1 2 P Sn Sm k 1 2 1 2 k n AMA528 (By Catherine Liu) Lecture 10 The Poisson Process 16/11/2011 10 / 15
  • Outline Exponential distribution Counting Process Poisson Process Nonhomogeneous PP Compound PPProperty 2Let X1 Xn i.i.d. U 0 t and the corresponding order statistics X 1 Xn.Then S1 Sn N t n X1 Xn .That is, the conditional joint pdf of S1 Sn given that N t n is n f s1 sn n 0 s1 sn t tn AMA528 (By Catherine Liu) Lecture 10 The Poisson Process 16/11/2011 11 / 15
  • Outline Exponential distribution Counting Process Poisson Process Nonhomogeneous PP Compound PPNonhomogeneous Poisson ProcessNonstationary Poisson proess: allow the arrival rate at time t to be a function of t.A counting process N t t 0 is said to be a nonhomogeneous Poisson Processwith intensity function t t 0, if 1 N 0 0; 2 The process has independent increments; 3 P N t h N t 1 t h o h ; 4 P N t h N t 2 o h . tLet m t 0 y dy. Then m t is called the mean value function of thenonhomogeneous Poisson process. And n m s t m s m s t m s P N s t N s n e n 0 n AMA528 (By Catherine Liu) Lecture 10 The Poisson Process 16/11/2011 12 / 15
  • Outline Exponential distribution Counting Process Poisson Process Nonhomogeneous PP Compound PPExample 2Eg 2 (example 5.20, pp.15): Siegbert runs a hot dog stand that opens at 8 am. From 8am until 11am customers seem to arrive, on the average, at a steadily increasing ratethat starts with an initial rate of 5 customers per hour at 8 am and reaches a maximumof 20 customers per hour at 11 am. From 11 am until 1 pm the (average) rate seemsto remain constant at 20 customers per hour. However, the (average) rate seems toremain constant at 20 customers per hour. However, the (average) arrival rate thendrops steadily from 1pm until closing time at 5pm at which time it has the value of 12customers per hour. If we assume that the numbers of customers arriving atSiegbert’s stand during disjoint time periods are independent, then what is a goodprobability model for the above? What is the probability that no customers arrivebetween 8:30 am and 9:30 am on Monday morning? What is the expected number ofarrivals in this period? AMA528 (By Catherine Liu) Lecture 10 The Poisson Process 16/11/2011 13 / 15
  • Outline Exponential distribution Counting Process Poisson Process Nonhomogeneous PP Compound PPCompound Poisson ProcessLet Yi i 1 be a family of i.i.d. r.v.s which are independent of a Poisson process N t t 0 . N tLet X t Yi t 0. Then the r.v. X t is said to be a compound Poisson r.v. and i 1the stochastic process X t t 0 is said to be a compound Poisson process.Eg: Let N(t) be # of customers leave a supermarket by time t distributed withPoisson t . Let Yi i 1 2 , the amount spent by the i-th customer, i.i.d. Let X t N tbe the total amount of money spent by time t. Then X t Yi t 0. And i 1 X t t 0 is a compound Poisson process.Remark: Let Yi 1, the X t N t , a usual Poisson process. X t X t N t N t Y1 t Y1 2 Var X t Var X t N t Var X t N t t Y1 AMA528 (By Catherine Liu) Lecture 10 The Poisson Process 16/11/2011 14 / 15
  • Outline Exponential distribution Counting Process Poisson Process Nonhomogeneous PP Compound PPExample 3Eg 3 (exampel 5.22, pp. 19) Suppose that families migrate to an area at a Poissonrate 2 per week. if the number of people in each family is independent and takeson values 1, 2, 3, 4 with respective probabilities 1 , 3 , 1 , 6 , then what is the expected 6 1 3 1value and variance of the number of individuals migrating to this area during a fixdfive-week period? AMA528 (By Catherine Liu) Lecture 10 The Poisson Process 16/11/2011 15 / 15