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The cost of ordering y units of product is a

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The cost of ordering y units of product is a specified function c(y) and it takes l units time until order is delivered. In addition, the shop pays an inventory holding cost of h per unit item per b unit time. Whenever customers demands more of the product than is presently available, than the amount on hand is sold and the remainder of the order is order is lost to the shop. Define variables and events to analyze this model and estimate the shop’s expected profit up to fixed time T. (10 marks) 6. Suppose that the different policyholders of a casually insurance company generate claims according to independent Poisson process with a common rate λ , and that each claim amount has distribution F. Suppose also that new customers sign up according to a Poisson process with rate v, and each existing policyholder pays the insurance
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firm at a fixed rate C per unit time. This system states with n customer and initial capital a 0, find then. Define the variable and enters for this system and estimate the probability that the firm’s capital is always non negative at all times up to time T. (20 marks) 7. A system needs n working machines to be operational. To guard against machine breakdown, additional machines are kept available as spares. Whenever a machine breaks down it is immediately replaced by a spare and is itself sent to the repair facility, which consists of a single repair person who repair failed machines one at a time. Once a failed machine has been repaired it becomes available as a spare to be used when the need arises. All repair times are independent random variables having the common distribution function G. Each time a machine is put into use the amount of time it functions before breaking down is a random variable, independent of the past, having distribution function f. The system is said to “Crash” when a machine fails and no spares are available. Define variables and events to analyze this system and write a model in simulating to estimate the time at which the system crashes. ( 2 0 m a r k s ) 8. Draw a flow diagram for simulating the Repair model. (10-marks) 9. The quantity X = which is the arithmetic average of the n data values, is called the sample mean. Proved that X, sample mean is an unbiased estimator of population mean θ . And also find its mean square error. (10 marks) 10. The quantity S 2 = 2 1 () 1 i i n X X n = is called the sample variance and show that sample variance S is an unbiased estimator of population variance σ . (10 marks) 11. For any set at number x , …. ., x prove algebraically that ( x – x ) = x - n x where x = x /n . (10 marks) 12. Write a method for determining when to stop generating new data to estimate a population mean. (10 marks) 13. Write a procedure for determining when to stop generating new values to estimate a probability. (The data values are Berroulli random variables). (10 marks) 14. If the first three data values are X1=5, X2=14, X3=9, and then find their sample mean and simple variance. (10 marks) 15. Suppose we are interested in estimating θ
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The cost of ordering y units of product is a specified...

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