errors every 1 hour unless he falls asleep, in which case he just reports the total errors

Transcription

1 I. First Definition of a Poisson Process A. Definition: Poisson process A Poisson Process {X(t), t 0} with intensity λ > 0 is a counting process with the following properties. INDEPENDENT INCREMENTS. For all t 0 = 0 < t < t < < t n, then X(t ) X(t 0 ), X(t ) X(t ),..., X(t n ) X(t n ) are independent random variables.. STATIONARY INCREMENTS with Poisson Distribution. For all s 0, t > 0, X(s + t) X(s) Poisson(λt). 3. X(0) = 0 B. Claim:. E [X(t)] = λt. Var [X(t)] = λt C. Verifying that a process is Poisson: verify three conditions of definition. Relatively easy. Can be verified with knowledge of process.. Difficult to verify Poisson distribution. 3. Easy. Counting begins at time t = 0. II. Example: You run a high-throughput center. Every hour, you run a control to make sure their is no excess of errors. Suppose the regular (acceptable) error rate is error per control run (so error per every hour). Your tech records the number of errors every hour unless he falls asleep, in which case he just reports the total errors accumulated when he wakes up. Suppose, your tech reports the following errors 9:00-9:30 0 9:30-0:00 3 0:00 - :00 7 What is the probability of observing such data if the true error rate is error per every hour? ( ( ) ) ( ( ) ) P (data) = P X = 0 P X = 3 P (X() = 7) [ ( e 0.5 = e 0.5 3! = e 3 [ 4 7! 3! ) 3 ] [e ] 7 7! ]

2 III. Second Definition of Poisson Process A. Definition: o(h) If f(h) is o(h), then lim h 0 f(h) h = 0. Little-o notation means that the function f(h) approaches 0 faster than h itself approaches 0. Examples. f(x) = x is o(h).. f(x) = x is not o(h). 3. If f(x) and g(x) are o(h), then f(x) + g(x) is o(h). 4. If f(x) is o(h), then cf(x) is o(h). 5. Therefore, all linear combinations of little-o functions are little-o. B. Definition : Poisson Process with rate λ.. The process has independent and stationary increments.. P [X(h) = ] = λh + o(h) 3. P [X(h) ] = o(h) 4. X(0) = 0 C. Theorem: The two definitions given are equivalent. (Proof not given.) D. Verification that a process is Poisson is now a bit easier. One need not confirm that the Poisson distribution applies, rather only that in any small interval the probability of event is linearly proportional to the rate and interval length and the probability of more than one event in a small interval is negligible. IV. Nonhomogeneous Poisson Process A. A nonhomogeneous Poisson Process is one in which the rate λ(t) is not constant throughout time t. In this case, the increments ( t ) X(t) X(s) Poisson λ(u)du and P (X(t + h) X(t) = ) = λ(t)h + o(h). Increments are still independent for any non-overlapping intervals, but they are no longer stationary. s

3 B. Technique to convert nonhomogeneous to homogeneous Poisson Process Nonhomogeneous Poisson process are useful but really pose very little problem because one can convert a nonhomogeneous process to a homogeneous process. To do this, we define a new way of measuring time. In this new time scale, denoted by s, the process is homogeneous. The conversion from regular time t to modified time s is accomplished through s = Λ(t) = t 0 λ(u)du, where λ(u) is the rate of the nonhomogeneous process. Then, the new process in the new time is Y (s) = X(t) and it {Y (s), s 0} is time homogeneous with rate. V. Law of Rare Events The law of rare events provides a way to connect the Poisson process assumptions to real-life. It also provides a means of showing that definition implies the Poisson distribution. A. Law of Rare Events The law of rare events describes the situation when there are many, many opportunities for an event to happen, but the probability of the event happening at any one of those opportunities is very small (think car accident at a busy intersection). A mathematical scenario is the binomial random variable X Bin(n, p) as n and p 0. If these limits occur with Np = µ constant, then in the limit. P (X = k) Poisson(µ) B. Law of Rare Events Connection to Poisson Process The Law of Rare events is connected to the Poisson process as follows. Imagine partitioning the time interval (0, t] into subintervals of length h. The number of arrivals in each subinterval is independent (by property ). The probability of a single event in any interval is λh + o(h) λh for small h (by property ). There will either be 0 or event in each subinterval as h 0 (by property 3). Let the number of intervals n = t/h as h 0. Then np = t λh = λt is constant so h the Poisson approximation with mean λt applies. 3

4 C. Law of Rare Events with Varying Probability Consider the Binomial distribution again, but suppose that the probability of success on the ith trial is p i, then if ξ i represents the outcome of the ith trial ( for success, 0 for failure), Y = ξ + ξ + + ξ n is the number of successes in the first n trials. We have P (Y = k) = ξ + +ξ n=k It turns out (proved in the book) that P (Y = k) µk e µ k! i p ξ i i ( p i) ξ i. n p i, where µ = i= p i. So again, we conclude that the probability of k successes can be approximated by a Poisson distribution, this time with mean equal to the sum of the probabilities. The approximation becomes very good as the p i 0, showing that the right-hand side goes to 0, bounding the difference between the approximation and the true probability of k successes very tightly. For nonhomogeneous Poisson processes, the probability of an arrival in time interval (u, u + h] is approximately λ(u)h and the mean of the approximating Poisson is i= λ(0)h + λ(t )h + + λ(t n )h t 0 λ(u)du, where 0 < t < t < < t n < t is a partition of the interval (0, t] into subintervals of length h and the limit is as h 0. VI. Cox Processes A. Definition: Consider a possibly nonhomogeneous Poisson-like process {X(t), t 0} where the time-variable rate {λ(t), t 0} is itself a stochastic process. Then, the resulting process X(t) is called a Cox process. B. The Cox Process does not necessarily have stationary increments nor independent increments. Hence, it is not a Poisson Process, but is a generalization of the Poisson Process. C. Motivation: The Poisson process examples we considered before can be viewed as Cox processes. For example, the number of people who have entered a store could be viewed as a Poisson process if there are many random factors that lead to a random, nonhomogeneous rate {λ(t), t 0}. 4

5 D. Example: Consider a Poisson-like process {X(t), t 0}, where the rate θ is a random variable with pdf f(θ). Then, the conditional process X(t) = x θ = λ is a regular Poisson process with rate λ, so P (X(t) = x θ = λ) are from the Poisson distribution. The unconditional distribution is P (X(t) = x) = 0 (θt) x e θt f(θ)dθ. x! In other words, this Cox process has some stationary, but non-poisson distribution for the increments. E. Simpler Example: Consider even a simpler example where the random rate variable is discrete-state stochastic process, so f(θ) is a pmf. A real-life example of this Cox process would be the number of failures of a component when the failed components are replaced with new components drawn from a bin with n different types of components, each with different lifelengths. The nth component makes up p n proportion of the components in the bin. We will examine the properties of the wait time until the first failure given the component was randomly drawn from the bin. Let F (t) be the cdf of this failure time distribution. Then, n F (t) = p j e λjt. Now, consider the failure rate r(t) = j= j= p jλ j e λ jt j= p je λ jt. Let λ i = min(λ,..., λ n ), then multiply top and bottom by e λ it to obtain r(t) = j= p jλ j e (λ j λ i )t j= p je (λ j λ i )t Now, take the limit as t, so = p iλ i + j i p jλ j e (λ j λ i )t p i + j i p jλ j e (λ j λ i )t. lim r(t) = p iλ i = λ i. t p i 5

6 We conclude that as the component ages, the failure rate converges to that of the component type with the smallest failure rate. In other words, if a component has survived so long, then it is very likely to be a component of the longest lasting type. We have only just touched on what one could do to analyze this process, considering only the time to first failure. VII. Important random times in a Poisson Process A. Definition: waiting time/arrival time of a Poisson Process Let W i be the waiting time for the ith event. It is simply the time of the ith occurrence of the event the Poisson Process is tracking. We define W 0 = 0. B. Definition: sojourn time/interarrival time of a Poisson Process Let S i = W i+ W i be the time between two arrivals of the event. S i is called the sojourn time. 6