EXERCISES

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that the total amount claimed during a given year has a negative binomial distribution with parameters / ln(1 ) and 1 120 An insurance company has two policies with xed remittances Claims from the policies 1 and 2 arrive according to independent Poisson processes with respective rates 1 and 2 Each claim from policy i is for a xed amount of ci , where c1 and c2 are positive integers Explain how to compute the probability distribution of the total amount claimed during a given time period 121 It is only possible to place orders for a certain product during a random time T which has an exponential distribution with mean 1/ Customers who wish to place an order for the product arrive according to a Poisson process with rate The amounts ordered by the customers are independent random variables D1 , D2 , having a common discrete distribution {aj , j = 1, 2, } (a) Verify that the mean m and the variance 2 of the total amount ordered during the random time T are given by 2 2 E(D1 ) and 2 = E(D1 ) + 2 E 2 (D1 ) (b) Let {pk } be the probability distribution of the total amount ordered during the random time T Argue that the pk can be recursively computed from m= pk = +

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starting with p0 = /( + ) 122 Consider a non-stationary Poisson arrival process with arrival rate function (t) It is assumed that (t) is continuous and bounded in t Let > 0 be any upper bound on the function (t) Prove that the arrival epochs of the non-stationary Poisson arrival process can be generated by the following procedure: (a) Generate arrival epochs of a Poisson process with rate (b) Thin out the arrival epochs by accepting an arrival occurring at epoch s with probability (s)/ and rejecting it otherwise 123 Customers arrive at an automatic teller machine in accordance with a non-stationary Poisson process From 8 am until 10 am customers arrive at a rate of 5 an hour Between 10 am and 2 pm the arrival rate steadily increases from 5 per hour at 10 am to 25 per hour at 2 pm From 2 pm to 8 pm the arrival rate steadily decreases from 25 per hour at 2 pm to 4 per hour at 8 pm Between 8 pm and midnight the arrival rate is 3 an hour and from midnight to 8 am the arrival rate is 1 per hour The amounts of money withdrawn by the customers are independent and identically distributed random variables with a mean of $100 and a standard deviation of $125 (a) What is the probability distribution of the number of customers withdrawing money during a 24-hour period (b) Calculate an approximation to the probability that the total withdrawal during 24 hours is more than $25 000 124 Parking-fee dodgers enter the parking lot of the University of Amsterdam according to a Poisson process with rate The parking lot has ample capacity Each fee dodger parks his/her car during an Erlang (2, ) distributed time It is university policy to inspect the parking lot every T time units, with T xed Each newly arrived fee dodger is ned What is the probability distribution of the number of fee dodgers who are ned at an inspection 125 Suppose customers arrive according to a non-stationary Poisson process with arrival rate function (t) Any newly arriving customer is marked as a type k customer with probability pk for k = 1, , L, independently of the other customers Prove that the customers of

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