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STAT4207/5207 Homework assignment 3 Due by March 31, 2021. Make sure you justify your answer. Write down complete derivations for full credits. Exercise 1 Poisson processes can also be defined for sets other than the positive half-line. For example, a homogeneous spatial Poisson process with rate λ on R2 is a random mechanism that generates points (the “events”) in the plane in such a way that the following two conditions hold: i) The number of points landing in disjoint regions are independent random variables; ii) The number of points landing in a region A ⊂ R2 has a Poisson distribution with mean λ× area(A). If we use the notation N(A) to denote the number of points in the region A ⊂ R2, we thus have N(A) ∼ Poisson ( λ× area(A) ) . Suppose water wells in a desert are distributed as a homogeneous spatial Poisson process with rate λ = 0.1 per square mile. Suppose you are somewhere in the desert desperate for water. a) Let R be the (Euclidean) distance from 0 (where you are) to the nearest well. Show that P(R > r) = e−λπr2 , r ≥ 0. b) Use your answer in a) to find E[R]. c) What is the probability that there is a well within 1 mile? Exercise 2 Let (N(t) : t ≥ 0) be a non-homogeneous Poisson process with intensity function λ(t) = { 2 if t is in (i, i+ 1] for i an even non-negative integer 4 if t is in (i, i+ 1] for i an odd non-negative integer a) What is the distribution of the number of arrivals in the time period (3.5, 5]? b) Calculate the probability that there are more than 2 arrivals in this time period. c) Given that T4 = 1.5, where T4 denotes the time of the fourth arrival, what is the probability that T5 > 6, where T5 denotes the time of the fifth arrival? Exercise 3 Electrons arrive at an anode in accordance with a Poisson process (N(t) : t ≥ 0) with constant rate λ. We shall denote Ti the time of the arrival of the ith electron. The ith electron produces a current with intensity Yi. We make the following simplifying assumptions: • The intensities Y1, Y2, . . . are i.i.d. with distribution function F , and are independent of the counting process (N(t) : t ≥ 0); • The intensities of the arriving electrons are assumed to decrease exponentially in time with rate ν. More specifically, at time t > Ti, the intensity of the ith electron, Yi, contributes Yie −ν(t−Ti) to the total intensity of the current at time t. 1 We can thus represent the total intensity of the current at time t by S(t) = N(t)∑ i=1 Yie −ν(t−Ti) . a) Show that S(t) has the same distribution as a compound Poisson process with parameters λt and Y e−νU , i.e., S(t) ∼ CPP (λt, Y e−νU ) where Y ∼ F and U ∼ Unif(0, t). b) Prove that E[W (t)] = λtE[X] and Var(W (t)) = λtE[X2] for an arbitrary compound Poisson process W (t) ∼ CPP (λt,X). c) Derive the expectation and variance of S(t) as t→∞. Hint for a): consider the conditioning theorem (Lecture 12/13). Exercise 4 A system consists of 5 components. The times to failure for these components are independent exponential random variables with parameter λ = 2. The system functions as long as at least 2 components are functioning. At time t = 0 all 5 components are functioning. Let T be the time to failure for the system. Find a) The probability density function of T b) P(T ≤ 0.1) c) Var(T ) Hint : consider theorem 6 of lecture 10 and the following lemma: Lemma 1. Let X1, X2, . . . , Xn with n ≥ 2, be independent exponential random variables with pair- wise distinct parameters λi. The density of Sn = ∑n i=1Xi is given by fSn(x) = [ n∏ j=1 λj ] n∑ i=1 e−λix∏n k 6=i k=1 (λk − λi) x > 0. Exercise 5 Consider an M/M/1 queue with waiting room capacity 3 (meaning that, when customer arrives and there are 4 customers already in the system, the customer is turned away). Let the arrival rate be λ = 4 and the service rate µ = 8. Let X(t) denote the number of customers in the system at time t. a) In the long run, what is the proportion of customers that is turned away? b) Find the embedded Markov Chain. c) Let τ1,4 = inf{t : X(t) = 4|X(0) = 1}. Find E[τ1,4]. d) Suppose that X(t) = 2. Find the probability that the next new state to be visited is 3, and the waiting time for this to occur exceeds 1. 2