Arrivals of customers at a novelty shop form a Poisson process with rate (per nour) that is uniformly distributed on (0,5). What is the expected number of arrivals during the first 3 hours?
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- Students arrive at Panda Express in the student center in accordance with a Poisson processwith rate λ. However, if the the student arriving finds n students already in line (including all who arewaiting and who are currently being served), then the student will enter the line with probability αn.Assume that there is only one person working at Panda Express and the time spent with each studentfollows an exponential distribution with rate μ. Set up this process as a birth and death process anddetermine the birth and death rates. Draw the rate diagram (state transition diagram).The number of requests for assistance received by a towing service is a poisson process with rate α = 2 per hour. After realizing that he did not miss any calls for assistance when taking a 15 minute break, the operator decides to take another 15-minute break. What is the probability that he will not miss any calls again? We want to determine P[no calls in time interval (0 , 1/2)│no calls in time interval (0 , 1/4)], where the unit used for time interval is hours.A traffic light is red for 20 seconds and green for 40 seconds. Cars arrive at the traffic light as a Poisson process with rate 2 per minute. They take exactly 20 seconds to go through the light. A car cannot start going through until the previous one has cleared the light. Assuming, as an initial condition, that there are no cars at the traffic light when it turns red: a. What is the probability that at least one car will arrive when the light is still red?b. What is the probability that there will be no cars at the light when the light next turns red?
- Taking expectation of a Wiener process wich is normally distrubuted with n(0,1) what is the value of that expectation?A radioactive mass emits particles according to a Poisson process. The probability that no particles will be emitted in a two-second period is 0.5. What is the probability that no particles are emitted in a four-second period?Stock of a branded product at a local shop is being replenished regularly throughout the day. Demands from customers are independent and each customer is allowed to purchase only one copy of the product. Demands from customers occur singly and at a rate of 2 per 30 minutes intervals. By applying the properties of exponential distribution, calculate the expected time when a stock of 9 such items need to be replenished.
- Pedestrians approach a crossing from the left and right sides following independent Poisson processes withaverage arrival rates of λL = 5 and λR = 1 arrivals per minute. Each pedestrian then waits until a lightis flashed, at which time all waiting pedestrians must cross to the opposite side (either from left to right orfrom right to left). Assume that the left and right arrival processes are independent, that the light flashesevery T = 2 minutes, and that crossing takes zero time – it is instantaneous.1. What is the probability that in a particular crossing, there are total 10 pedestrian and they are allcrossing from left to right?X1, X2 are independent continuous random variables with a uniform distribution in the interval (a, b). Calculate the moment generating function, expected value, variance of random variable Y = X1 + X2The arrival of customers to a small local food restaurant may be modeled by a Poisson process with rate of 1 per 15 min period. Assume customer will only arrive if there is at least one customer present. Each customer on average stays for a time exponentially distributed with mean 20 minutes. This is modeled as a birth-death process. (i) Compute the probability that the capacity of 20 will be reached if we start with 5 customers.
- An entrepreneur opened a small hardware store in a street mall. During the first few weeks, business was slow, with the store averaging only one customer arrived every 15 minutes. Assume that the random arrival of customers is Poisson distributed. What is the probability that at least two customers arrive during a one-hour period?Occurrences of landfalling hurricanes during an El Niño event are modeled asa Poisson process in Bove et al. (1998). The authors assert that “During an ElNiño year, the probability of two or more hurricanes making landfall in theUnited States is 28%.” Find the rate of the Poisson process.A hen lays N eggs, where N has a Poisson distribution with parameter λ. Each egg hatches independently with probability p. a.) Calculate the marginal distribution of X, where X is the number of successful hatches. b.) Calculate the conditional expectation E(N|X).