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# Let c be the speed of light in a vacuum. Suppose that c is unknown, and scientists wish ISBN: 9781466575578 423

## Solution for problem 12 Chapter 6

Introduction to Probability | 1st Edition

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Problem 12

Let c be the speed of light in a vacuum. Suppose that c is unknown, and scientists wish to estimate it. But even more so than that, they wish to estimate c2, for use in the famous equation E = mc2. Through careful experiments, the scientists obtain n i.i.d. measurements X1, X2,...,Xn N (c, 2). Using these data, there are various possible ways to estimate c2. Two natural ways are: (1) estimate c using the average of the Xj s and then square the estimated c, and (2) average the X2 j s. So let Xn = 1 n Xn j=1 Xj , and consider the two estimators T1 = X 2 n and T2 = 1 n Xn j=1 X2 j . Note that T1 is the square of the first sample moment and T2 is the second sample moment. (a) Find P(T1 < T2). Hint: Start by comparing ( 1 n Pn j=1 xj ) 2 and 1 n Pn j=1 x2 j when x1,...,xn are numbers, by considering a discrete r.v. whose possible values are x1,...,xn. (b) When an r.v. T is used to estimate an unknown parameter , the bias of the estimator T is defined to be E(T) . Find the bias of T1 and the bias of T2. Hint: First find the distribution of Xn. In general, for finding E(Y 2) for an r.v. Y , it is often useful to write it as E(Y 2) = Var(Y )+(EY)2

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STA3024 Notes 1/6/16 *posted course notes: incomplete notes to be filled in during lecture* 5 assignments: 20 points each -solutions uploaded the night the assignment is due -no late work permitted -due in class Extra credit will be offered at the end of the semester: 10 points First two exams are noncumulative, the third one is cumulative -2/3 of...

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##### ISBN: 9781466575578

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