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- A company is considering a project which has three options. The payoffs are shown in the body of thetable. In this instance, the payoffs are in terms of present values, which represent equivalent currentdollar values of expected future income less costs. This is a convenient measure because it places allalternatives on a comparable basis. If a small facility is built, the payoff will be the same for all threepossible states of nature. For a medium facility, low demand will have a present value of $7 million,whereas both moderate and high demand will have present values of $12 million. A large facility willhave a loss of $4 million if demand is low, a present value of $2 million if demand is moderate,and a present value of $16 million if demand is high.The payoff table depicting revenues and is shown in the following table. (a) Which alternative should the manager choose under the maximax criterion? (b) Which option should the manager choose under the maximin criterion? (c) Which option…1. If there are n random variables such that each expectation exist then prove that E(X1+…….+Xn ) = E(X1)+……..+E(Xn). Assume discrete.What is the simple random walk on integers? Does it always return to the origin if p = ½ ? if p ≠ ½?
- In the B&K model of Example 18.5-1, suppose that the interarrival time at the checkout area is exponential with mean 5 minutes and that the checkout time per customer is also exponential with mean 10 minutes. Suppose further that will add a fourth counter. Counters 1,2, and 3 will open based on increments of two customers and counter 4 will open when there are 7 or more in the store. (a) The steady-state probabilities, for all . (b) The probability that a fourth counter will be needed. (c) The average number of idle counters.A city uses three pumps to carry water from a river to a reservoir. Pumps A and B are new, and have a probability of failing of 0.015 on any day. Pump C is older, and has a probability of failure of 0.08 on any day. Pumps A and B operate Monday-Friday. On Saturday, pumps A and C operate while pump B is serviced. On Sunday, pumps B and C operate while pump A is serviced. Answer the following questions, assuming that the pumps operate independently of one another, and independently from day to day. Determine the probability that pump A works on a day that it is in use. Find the probability that pumps A and B both fail on a day they are both in use. Compute the probability that at least one pump fails on any Sunday. Find the probability that pump A works, and C fails on any Saturday. Determine the probability that no pumps fail in a week.Everyday, James reverses his car from his driveway on to the road in such a way that there is a very small probability P that his car will be involved in a collision. ( i) Show that the probability that there will be no collision in a five-day week is (1−P)^5 and state one assumption that is made in your answer. (iii) Let P = 0.001. Check that the Poisson approximation can be used, and find the approximate probability that James will avoid a collision in 500 days.
- Suppose that the probability of observing |0⟩ in the state |ϕ1⟩ is 1/4 and the probability ofobserving |1⟩ in the qubit |ϕ2⟩ is 1/3.A group of Sports Science students (n = 20) are selected from the population to investigate whether a 12-week plyometric-training programmed improves their standing long jump performance. In order to test whether this training improves performance, the students are tested for their long jump performance before they undertake a plyometric-training programme and then again at the end of the programmed .The following table present the results of before and after the training programmed. Test the claim that their long jump performance is higher after the training. Student Number (Before the training) Jump1 (After the training) Jump2 1 2.25 2.24 2 2.42 2.48 3 2.26 2.29 4 2.58 2.62 5 2.62 2.64 6 2.16 2.18 7 2.40 2.44 8 2.62 2.67 9 2.35 2.39 10 2.44 2.47 Follow the steps in hypothesis testing. Answer the following. What is the appropriate statistical test to use?…For ALL the following statements, evaluate each statement as either TRUE or FALSE. Then, justify your answer with a careful explanation. Please note that explanations may also involve mathematical and/or graphical illustrations. If all the Gauss-Markov assumptions of the OLS estimator hold, with the exception that the error term is heteroskedastic, then the OLS estimator is biased.