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Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2008)Problem Set 12Due: NeverTopics: Classical Parameter Estimation, Classical Linear Regression, ClassicalHypothesis TestingAlthough this problem set is never due, you are strongly encouraged to go through the problems andattempt to solve them. You will be tested on this material in the final exam. Solutions will be postednext week.1. Given the five data pairs (xi,yi) in the table below,x 0.8 2.5 5 7.3 9.1y -2.3 20.9 103.5 215.8 334we want to construct a model relating x and y.WeconsideralinearmodelYi= θ0+ θ1xi+ Wi,i=1, ···, 5,and a quadratic modelYi= β0+ β1x2i+ Vi,i=1, ···, 5.where Wiand Virepresent additive noise terms, modeled by independent normal random variableswith mean zero and variance σ21and σ22, respectively.(a) Find the ML estimates of the linear model parameters.(b) Find the ML estimates of the quadratic model parameters.(c) Assume that the two estimated models are equally likely to be true , and that the noiseterms Wiand Vihave the same variance: σ21= σ22. Use the MAP rule to choose betweenthe two models.2. Let f be the fraction of a very large (consider it infinite) population that has a particularattribute. As an estimator for f ,weuseFn=Kn, where K is the number of people in a randomlychosen sample of size n that have the attribute of interest.(a) In terms of sample size n and parameter f, what are the expectation and variance of ourestimator for f?(b) In terms of f, find the smallest value of n (n>1) such that the standard deviation of Fnis less than or equal to 10−3. (Be conservative, and give the smallest value of n for everypossible value of f.)(c) Fully explain your answers to the following questions. Is the estimator Fn:i. unbiased?ii. consistent?iii. the maximum likelihood estimator?(d) Chebyshev’s inequality has been used to compute n, an adequate sample size to satisfyP(|f − Fn|≥) ≤ 0.10 for every possible value of f. The result of that calculation isn = 1000.Compiled December 4, 2008 Page 1 of 3Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2008)i. What value of  was used in that calculation?ii. Using your result from part (i), evaluate the CLT-based approximation to n that satisfiesthe same condition, P(|f − Fn|≥) ≤ 0.10.(e) For the case n = 400 and K = 100, specify your 95% confidence bounds for the estimatorfor f. The confidence intervals should be valid for every possible value of f.3. Consider a Bernoulli process X1,X2,X3, ... with unknown probability of success q. The kth inter-arrival time Tkis defined as:T1= Y1,Tk= Yk− Yk−1,k=2, 3, ...where Ykis the time of the kth success. Assume that q is the value of a random variable Q whichis uniformly distributed in the interval [0, 1] . In your computations you may find the followingequality useful:10qk(1 − q)mdq =k!m!(k + m +1)!(a) Compute the maximum a posteriori (MAP) estimate of Q given the first k recordings,T1= t1, ..., Tk= tk.(b) Compute the Least Mean Squares (LMS) estimate of Q from the first observation T1= t1.(c) Compare your answers in parts (a) and (b) with the coin flipping problem (Problem 2, parts(a) and (b) respectively) of recitation 20. Are your answers consistent?(d) For this part only, assume that Q is uniform in the interval [0.5, 1] . Find the linear leastsquares estimate (LLSE) of the second inter-arrival time (T2), from the observed first arrivaltime (T1= t1).(e) For this part, assume that q is an unknown parameter in the interval (0, 1]. Denote the trueparameter by q∗. ComputeQk, the maximum likelihood estimate (MLE) of Q given the firstk observations, T1= t1, ..., Tk= tk.4. (Depends on Lecture Material presented on 12/8) Under hypothesis H0, the PDF forT isCompiled December 4, 2008 Page 2 of 3Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2008)Suppose we are told that under an alternative hypothesis H1, the PDF for T is uniformly dis-tributed between 0 and 1, i.e.fT(t; H1)= 1 , 0 ≤ t ≤ 10 , otherwise(a) Compute the likelihood ratio function and approximately sketch the Operating Character-istics curve for all the decision rules that can be expressed as Likelihood Ratio Tests. (Putα = P (g(X)=H1; H0)onx-axis,and1− β = P (g(X)=H1; H1) on y-axis).(b) Determine an acceptance region for H0such that α, the probability of false rejection ofH0is equal to .04, and β, the probability of false acceptance of H0(when H1is the truehypothesis) is minimized subject to this condition. Find the minimum value of β.This test is to be based on a single experimental value of t.5. (Depends on Lecture Material presented on 12/8) Let N(µ, σ2) denote a Gaussian PDFwith expectation µ and variance σ2. We use a single experimental value of Q for a hypothesistest between H0and H1, which are:H0: Q is N(0, 16) H1: Q is N(0, 9)The test is to be designed to minimize the probability of an incorrect conclusion.(a) With P (H0)=1/4, specify the H0acceptance region for statistic Q.(b) What is the smallest value of P (H0), if any, such that the resulting value test accepts H0for all values of statistic Q.6. (Depends on Lecture Material presented on 12/8) Assume:The weights of California oranges are independent experimental values of a Gaussian randomvariable with a standard deviation equal to 2 ounces.The weights of Florida oranges are independent experimental values of a Gaussian randomvariable with a standard deviation of 3 ounces.From a sample of n = 64 California oranges, we find the average of their weights to be 12 ounces.From a sample of m = 36 Florida oranges, we find the average of their weights to be 12.2 ounces.(a) Using a statistic which is to be the difference in the average of the weights in the twosamples, we wish to perform a significance test on the hypothesis that the distributionsfor the weights of California and Florida averages have the same expectation. What isthe critical value of α, the level of significance, at which we would be indifferent betweenaccepting and rejecting our hypothesis?(b) Do something reasonable to estimate the difference in the expected weights of Californiaand Florida oranges. “Interview” your estimator.(c) How much would we be concerned if the weights of


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MIT 6 041 - Problem Set 12

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