Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Survey

Document related concepts

Transcript

The Poisson Process Stat 6402 Counting process A stochastic process {N(t) : t ≥ 0} is a counting process if N(t) represents the total number of “events” that occur by time t. eg, # of persons entering a store before time t, # of people who were born by time t, # of goals a soccer player scores by time t. N(t) should satisfy: N(t)>0 N(t) is integer valued If s<t, then N(s)< N (t) For s<t, N(t)-N(s) equals the number of events that occur in (s, t] http://www.ms.uky.edu/~mai/java/stat/countpro.html Independent and stationary increments characterization of the Poisson process The counting process {N(t), t>0} is said to be a Poisson process having rate λ if 1. N (0) = 0 2. The process has independent increments. A counting process is said to have independent increments if for any non-overlapping intervals of time, I1 and I2, N(I1) and N (I2) are independent. 3. The number of events in any interval of length t is Poisson with mean λt. A counting process is said to have stationary increments if for all t>0, s>0, N(t+s)- N(s) has a distribution that only depends on t, the length of the time interval. Here, P{N(t+s)- N(s)=n} = e−λt(λt)n/n!, n = 0, 1, … E[N(t)] = λt Little o(t) results Let o(t) denote any function of t that satisfies o(t)/t → 0 as t → 0. Examples include o(t) =tn, n>1 For a Poisson process, P(N(t) = 1) = λt + o(t) P(N(t) > 1) = o(t). Interarrival and waiting time distribution {Tn, n= 1, 2, …} is called the sequence of interarrival times. P{T1>t} = P{N(t)=0} = e-λt P{T2>t | T1 = s} = P{0 events in (s, s+t] | T1 = s} = P{0 events in (s, s+t] } = e-λt Tn are iid exponential with mean 1/λ Sn is the arrival time of the nth event, also called the waiting time until the nth event. Sn = T1+ T2+…. +Tn Sn has a gamma distribution with parameters n and λ Partitioning a Poisson random variable Suppose each event is classified as a type I event with probability p or a type II event with probability 1-p. If N(t) ∼ Poiss(α) and if each object of X is, independently, type 1 or type 2 with probability p and q = 1− p, then in fact N1(t) ∼ Poiss(pα), N2(t) ∼ Poiss(qα) and they are independent. Suppose there are k possible types of events, represented by Ni(t), i=1, …k, then they are independent Poisson random t variables having means E[Ni(t)] = λ Pi(s)ds 0 Conditional distribution of the arrival times Theorem: Given that N(t)=n, the n arrival times S1, S2,…Sn have the same distribution as the order statistics corresponding to n independent random variables uniformly distributed on the interval (0, t). f (S1,… Sn | n) = n!/ tn, 0<S1…<Sn Proposition : Given that Sn=t, the set S1,…Sn-1 has the distribution of a set of n-1 independent uniform (0,t) random variables. Compound Poisson process A stochastic process {X(t), t>0} is said to be a compound Poisson process if it can be represented as X(t) = N (t ) Yi i 1 Here {N(t), t>0} is a Poisson process, and {Yi, i>0} is a family of independent and identically distributed random variables that is also independent of N(t). eg. Suppose customers leave a supermarket with a Poisson process. Yi represents the amount spent by the ith customer and they are iid. Then {X(t), t>0} is a compound Poisson, X(t) denotes the total amount of money spent by time t. Compound Poisson process E[x(t)] = λt E[Yi] Var[x(t)]=λt E[Yi 2] Suppose that families migrate to an area at a Poisson rate 2 per week. If the number of people in each family is independent and takes on the values 1, 2, 3, 4 with respective probabilities 1/6, 1/3, 1/3, 1/6, then what is the expected value and variance of the number of individuals migrating to this area during a fixed five-week period?