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- 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)?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?Assume that the variables Y1, Y2,... in a compound Poisson process have Bernoulli distribution with parameter p . Show that the process reduces to the Poisson process of parameter λp.
- Starting at time 0, amusement park attendees enter the park and arrive according to a Poisson process with rate equal to 48 per hour. If 10 attendees arrive in the first minute, what is the expected number of attendees that arrive in the first two minutes?Suppose that 2% of the employed lose their job each month. Further suppose that 25% of the unemployed find a job each month. a. What is the average duration of employment in this labor market? b. Do you think this is a reasonable figure given estimates of the natural rate? Why or why not? c. If you wanted to reduce the steady-state level of unemployment, what type of policies might you pursue and why?Suppose we know that the number visits to a webpage can be modeled as a Poisson Process with rate α = 4 per minute. The probability that the webpage gets exactly 10 visits during a particular 2 minute period is approximately
- A small bank has two tellers, one for deposis and one for withdrawals. The service time for each teller is exponentially distributed, with a mean of 1 min. Customers arriving at the bank according to a poisson process, with mean rate 40 per hour; it is assumed that depositors and withdrawers constitute separate poisson processes, each with mean rate 20 per hour , and that no customer is both a depositor and a withdrawer. The bank is thinking of changing the current arrangement to allow each teller to handle both deposits and withdrawals. The bank would expect that each teller's mean service time would increase to 1.2 min, but it hopes that the new arrangement would prevent long lines in front of one teller while the other teller is idle, a situation that occurs from time to time under the current setup. Analyze the two arrangements with respect to the average idle time of teller and the expected number of customers in the bank at any given time.Suppose that the number of customers arriving to a store follows the Poisson process with rate 1/λ per minute. Choose an option that is not correct. The expected number of customers arriving to the store between 9am and 11am is twice bigger than the expected number of customers arriving to the store between 10am and 11am. The number of customers arriving to the store between 9am and 10am is independent of the number of customers arriving to the store between 10am and 11am. The expected number of customers arriving to the store between 9am and 10am is 60λ. The number of customers arriving to the store between 9am and 10am follows the same distribution as the number of customers arriving to the store between 10am and 11amTwo Poisson processes with rates λ1=6.507and λ2=0.368 resulted from splitting a Poisson process. What is the probability that an arrival from the "parent" process will belong to the second process with rate λ2? Specify your answer to four decimal places.
- Suppose that you would like to create composite indexes from three stocks: stk1, stk2, stk3. Their stock prices (Pt) and total shares outstanding (Qt) from day 0 to day 1 are shown as follows: stk3 splits two-for-one in day 1. P0 Q0 P1 Q1 stk1 40 250 50 250 stk2 50 100 50 100 stk3 60 150 50 300 Which answer is the closest value to the rate of return on a price-weighted index of the three stocks? A. 20% B. 25% C. 30% D. 35%a man is investigating the populaion of bear in two areas. Area 1 and Area 2. He expect the number of bear to be X and Y in area 1 and area 2 to be Poisson- distributeted. He expect the number og bear to be λ1 = 3 in area 1 and λ2 = 5 in area 2. Find P(X = 2) and P(X ≥3) and find an approximate value expression for P(X = Y)Suppose taxicab arrivals from 3 to 5 p.m. are a Poisson process. If the cabs arrive at an average rate of 18 per hour, what is the probability that a person will have to wait up to 5 minutes for a cab?
