A random-number generator simulates realizations from a Unif(0, 1) distribution. (a). distribution. Find a transformation h to transform the realizations to follow a Binomial(6,0.3) (b). the N(0, 1) CDF Table. Describe a strategy to transform the realizations to follow a N(5,9) distribution, using
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- Suppose that only 0.1% of all computers of a certain type experience CPU failure during the warranty period. Consider a sample of 4,000 computers.a. Can you use Poisson distribution as approximation for binomial? Why? b. Compute P(4<=X<=6) (Use the formula of the pmf).Find the moment-generating function of the negativebinomial distribution by making use of the fact that if k independent random variables have geometric distribu-tions with the same parameter θ, their sum is a random variable having the negative binomial distribution withthe parameters θ and k.Suppose that the number of accidents occurring on a highway each day is a Poisson random variable with parameter λ=3.a) Find the probability that 3 or more accidents occur today.b) Repeat part (a) under the assumption that at least 1 accident occurs today.
- ~if calls to your cell phone are a Poisson process with a constant rate lambda=2 calls per hour, what's the probability that, if you forget to turn your phone off in a 1.5 hour movie, your phone rings during that time? ~How many phone calls do you expect to get during the movie?Show that if the limiting conditions n→q, θ →0, while nθ remains constant, are applied to the moment-generating function of the binomial distribution, we get the moment-generating function of the Poissondistribution.Suppose that historically, 36% of applicants that are offered admittance to Georgia Southern actually enroll, while the others take offers somewhere else. If Georgia Southern will accept 9150 this coming year, what is the probability that less than 3250 will actually enroll? Use the normal approximation to the binomial. P(X < 3250) =
- A traffic enforcer counts the nunber of vehicles at a specific locationin the highway. In his counting, there are 420 vehicles arriving in a span of 1 hour. Assuming that the arrival of vehicles is poisson distributed, estimate the probabilities of having 7 vehicles arriving over a 30-second time interval, over the next 1 minute.The number of flaws in a certain brand of copper wire is modelled by a Poisson process. Suppose thaton average there is one flaw in 10 meters of wire. To make an electrical equipment, an engineer needs 5 pieces of flawless wires, each piece withlength one meter. Suppose that he orders 6 meters. What is the probability that he has to orderagain?20% of the workers in a factory work in plant A and 80% work in plant B. The number workers calling in sick per week follow a Poisson probability mass function with parameter λ=3 for plant A, and λ=5 for plant B. What is the probability that next week there will be 2 workers calling in sick?
- A large portfolio of policies is such that a proportion of p, 0<p<1 incurred claims during the last calendar year. An investigation examines a randomly selected group of 25 policies from the portfolio. Use a Poisson approximation to the binomial to calculate the approximate value for the probability that are at most four policies with claim in the case where p=0.1 and p=0.2. Comment briefly on the above approximations given that the exact values of the probabilities are 0.9020 and 0.4207.Suppose that earthquakes occur in a certain region of California, in accordance with a Poisson process, at a rate of seven per year. Assume that all required conditions for the Poisson process hold. a. What is the probability that in exactly three of the next eight years no earthquakes will occur? b. Approximate the probability that at most 3 of the next 30 years no earthquakes will occur by using an appropriate Poisson distribution.Suppose surveys by ABC team and XYZ team indicate that 44% of voters support the secession of company. Suppose further that a intern takes a sample of 20 random voters. 1. Using Bernoulli random variable, construct the probability distribution for each possible outcome on the number of independent company supporters out of the 20 individuals using the binomial formula and solve for the first and second moment. Expand the table as necessary. 2. Draw the PDF/PMF graph and label the first moment