(1)Phone calls in a customer service center arrive according to a Poisson process with rate of 6 calls per minute. For each call, the probability that it is a complaint is given by a Bernoulli distribution with parameter 0.3. Find (i) the expected number of complaints received in any given one hour period
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- The number of work-related injuries per month in a manufacturing plant is known to follow a Poisson distribution, with a mean of 4.0 work-related injuries a month. What is the probability that in a given month, no work-related injuries occur? That at least one work-related injury occurs?E The number of accidents occurring in a plant in a month follows Poisson distribution with mean as 5.2. The probability of occurrence of less than 2 accidents in the plant during a randomly selected month isA petrol pump station has 4 pumps. The service times follow the exponential distribution with a mean of 6 minutes and cars arrive for service in a Poisson process at the rate of 30 cars per hour. (i) What is the probability that an arrival would have to wait in line?(ii) Find the average waiting time, the average time a car spent in the system and the average number of cars in the system. (iii) For what percentage of time would a pump be idle an an average?
- The number of traffic accidents at a certain intersection is thought to be well modeled by a Poisson process with a mean of 3 accidents per year. Find the probability that less than three months elapse between accidents.The average number of collisions in a week during the summer months at a particular intersection is 2. Assume that the requirements of the Poisson distribution are satisfied. What is the probability that there will be exactly one collision in a week?Willow Brook National Bank operates a drive-up teller window that allows customers to complete bank transactions without getting out of their cars. On weekday mornings, arrivals to the drive-up teller window occur at random. Assume that Poisson probability distribution with an arrival rate of 24 customers per hour or 0.4 customer per minute can be used to describe the arrival pattern. Assume further that the service times for the drive-up teller follow an exponential probability distribution with a service rate of 36 customers per hour, or 0.6 customer per minute a.Use the single-channel drive-up bank teller operation to determine the average arrival time in minutes of customers b.Use the single-channel drive-up bank teller operation to determine the average service time in minutes of the drive-up teller.
- In a specific area, this period of time, it is believed that the number of thunderbolts that are falling is distributed according to the Poisson distribution with a rate of 75 thunderbolts per hour. In the next questions give your answer as a decimal number correct in three decimal places. During a specific minute time, find the probability that: (i) there are no thunderbolts in the area.(ii) at least one thunderbolt hits the area.Person P is walking with his dog every day. He decides to model the time intervals between dog encounters (i.e., how long does it take between two encounters) with the independent random variables Y₁, . . ., Yn . Number of dogs encounters per day, the number of customers entering a store per hour, the number of bacteria per liter of water, etc. are classically modelled using the Poisson distribution. With an exponential distribution, on the other hand, is used for modelling intervals of between events and how long a light bulb lasts, for example. After learning this, person P decides to model the observations in such a way that the corresponding random variables Y1,..., Yn for a random sample, that is, independent observation, of the exponential distribution Exp(1/µ), µ > 0. The parameter µ > 0 is used as the statistical model parameter 1 which is the expected value of each observation random variable Yi. (a) Form the maximum likelihood estimate μ^(y) in the model for the parameter μ…The number of bankruptcies filed in the district court has a Poisson distribution with an average of 6 per week. (a) What is the probability that there will be no bankruptcy filings during a given week? (Round your answer to three decimal places.) (b) What is the probability that there will be at least one bankruptcy filing during a given week? (Round your answer to three decimal places.) (c) Within what limits does Tchebysheff's Theorem suggest you would expect to see the number of bankruptcy filings per week at least 88.88% of the time? (Round your answer up to the nearest whole number.)
- An airport has 3 security lines, with each line capable of scanning 290 people per hour. People arrive in front of the security line at the rate of 850 per hour. Make the standard assumption of a Poisson distribution for arrivals and an exponential distribution for service times, and calculate the following: What is the probability that there are no people in the waiting line? Answer is NOT 0.02. What is the average length (number of people) in the waiting line? Answer is NOT 41.52.A manufacturing company claims that the number of machine breakdowns follows a Poisson distribution with a mean of two breakdowns every 500 hours. Let x denote the time (in hours0 between successive breakdowns. assuming that the manufacturing company's claim is true, find the probability that the time between successive breakdowns is at most five hours.The number of customers arriving per hour at a certain automobile service facility is assumed to follow a Poisson distribution with mean, = 7i. What is the mean number of arrivals daily if the facility operates for 8 hours perday? ii. Compute the probability that more than 10 customers will arrive in a 2-hour period.