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- Assume that N(t) is a Poisson process with rate λ=2024. ComputeP(N(s) =1000,N(t) =2023),for any s≤tLet {Xt : t > 0} be a compound Poisson process of parameter λ. Suppose that each of the summands of this process is constant equal to k ∈ IN . Find the distribution of Xt.Suppose the customers arrive at a Poisson rate of on eper every 12 minutes, and that the service time is exponential at a rate of one service per 8 minutes. What are the average number of customers in the system(L) and the average time a customer spends in the system(W)?
- Let {N_1(t)} and {N_2(t)} be two independent Poisson processes with rates λ1=1 and λ2=2, respectively. Find the probability that the second arrival in N_1(t) occurs before the third arrival in N_2(t). Round answer to 4 decimals.Use table D, the Standard Ultimate Survival Model with i=5%. Consider a 25-year term insurance isued to a life aged 35 with annual premiums payable throught the policy term, with sum insured $100,000 payable at the end of the year of death if death occurs during the term. Expenses are as follows: 10% of the 1st annual premium and 2% of each subsequent premium, with an inception expense of $100 and renewal expenses of (at the time of the payment of the 2nd and each subsequent premium) of $15. Calculate the annual premium (using the equivalence principle).The manager of a market can hire either Mary or Alice. Mary, who gives you service at an exponential rate 20 customers per hour, can be hired at a rate of $3 per hour. Alice, who gives service at an exponential rate of 30 customers per hour, can hired at a rate of $C per hour. The manager estimates that, on the average, each customer’s time is worth $1 per hour and should be accounted for in the model. Assume customers arrive at a Poisson rate of 10 per hour. a) What is the average cost per hour if Mary is hired? If Alice is hired? b) Find C if the average cost per hour is the same for Mary and Alice.
- If X1, X2, ... , Xn constitute a random sample of size n from an exponential population, show that X is a consis-tent estimator of the parameter θ.Determine the set of prices that would satisfy the condition of each the three markets using Gauss-Jordan method. Given following below in the picture.Give the resulting rate when two independent Poisson processes with rates λ1=2.546 and λ2=3.326 are merged. What is the exact rate?
- Consider a stable single-station queue. Suppose it costs $c to hold a customer in the system for one unit of time. Show that the long-run holding cost rate is given by cL, where L is the mean number of customers in steady state.In the machine-repair model, assume that there are 2 repairman (each working at the same rate) and 3 machines. (a) Draw a rate diagram and determine the steady-state distribution if ! = 4 and µ = 8 . (b) Evaluate the expected total cost per hour if each repairman costs $10 per hour and broken machines cost $5 per hourLet X1,X2,... be a sequence of identically distributed random variables with E|X1|<∞ and let Yn = n−1max1≤i≤n|Xi|. Show that limnE(Yn) = 0