# Variance s is an unbiased estimator of population

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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 (F) =E[X] by using the sample mean 1 n i i x X n = = . If the observed data are x i , i=1,….n, then the empirical distribution F e puts weight 1/n on each of the points x 1 ,…..., x n (combining weight of i=1 n i n X 1 n n 2 n i= 1 2 2 i= 1 i= 1 n i i i

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the x i are not all distinct). Prove that the bootstrap approximation to the mean square error MSE(F) is identical to the usual estimate of the mean square error.(10 marks) 16. A service system in which no new customers are allowed to enter after 5 Pm. Suppose that each day follow the same probability law and estimate the long-sum average amount of time a customer spends in the system. And then prove that the MSE of estimate of average time a customer spends in the system is identical to the bootstrap approximation. (20 marks) 17. Give a probabilistic proof of = = = n 1 i n 1 i 2 2 i 2 i nx x ) x (x where = = n 1 i i n x x letting X denote a random variable that is equally likely to take an any of the values x 1 ,…..,x n and then by applying the identify Var (X) = E[X ] – (E[X]) 2 . (10 marks) 18 . To estimate E[X], X1,……,X16 have been simulated with the following values resulting, 10,11,10.5,11.5,14,8,13,6,15,10,11.5,10.5,12,8,16,5. Based on these data, if we want the standard deviation of the estimator of E[X] to be less than 0.1, roughly how many additional simulation runs will be needed? (10 marks) 19. Suppose we want to use simulation to estimate 1 0 u x E e e dx θ = = . Using antithetic variables, discuss what kind of improvement is possible. (10 marks) 20. Consider a sequence of random numbers and let N be the first one that is greater than its immediate predecessor. That is N= min (n: n 2 , U n >U n-1 ). Explain what kind of improvement is possible on the variance of estimator “e” by using antithetic variable. (20 marks) 21. Suppose we were interested in using simulation to compute θ =E [e u ]. Here, a natural variant to use as a control is the random number U. Discuss what sort of improvement over the raw estimator is possible. (20 marks)