final--exam - IS my FINAL EXAM: STATISTICS 426 Moulinath...

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Unformatted text preview: IS my FINAL EXAM: STATISTICS 426 Moulinath Banerjee University of Michigan April 19, 2006 Announcement: The exam carries 70 points but the maximum you can score is 60. Half of your score contributes to your grade in the-course. (1) (2) (3) (4) Let X1 and X2 be i.i.d. N(O, 1) random variables. Define Yl = X1 and Y2 = XI/Xz. Compute the joint-distribution of (Y1, Y2) and Show that the marginal density of Y2 is: fY2(312)=7r( 1 , yzE(—~oo,oo). 1 + (10 points) Let X be a single random variable following EXpOQ. Define T(X) ’z 1 if X > 1 and T(X) = 0 otherwise. Set 1,!)(A) = e‘A. Show that T(X) is unbiased for $0) and find the information bound fer unbiased estimators of $0). Show that the variance of T(X) is strictly larger than the information bound. (You may use the fact that eA — 1 > A2.) (10 points) Suppose that X1,X2, . . . ,‘Xn are i.i.d. random variables following a geometric distribution. Thus, for each i, P(Xi = x) = «gm—1p where x is a positive integer, and 0 < p,q < 1 and p+q=L Compute an explicit expression for the probability that the minimum of the Xi’s is larger than a fixed integer as. What happens to this probability for fixed a: (say :1: 2 1) as it becomes large? What is the intuitive explanation behind this phenomenon? (10 points) A biologist is interested in measuring the ratio of mean weights of animals of two species. However, the species are extremely rare and after much effort she succeeds in measuring the weights of one animal from the first species and one from the second. Let X1 and X2 denote these weights. It is assumed that X; m N(91, 1) and X2 N N (62, 1). Interest lies in estimating 191/82. Compute the distribution of 92X1 —~ 6?ng «a? + 6% X1 — (81/92)){2 V(‘91/92)2 + 1 is a pivot. Discuss how you can use this pivot to construct a confidence set for the ratio of mean weights. (10 points) h(X17-X2561162) = and conclude that (5) Consider two particles situated at locations X1 and X2 on the horizontal axis and particles (6) Y1 and Y2 situated at locationaYl and Y5; on the vertical axis. It may be assumed that all random variables are independent. Furthermore X1. and X2 are i.i.d. N (0,0?) and Y1 and Yg are i.i.d. N(O,o§). You only observe X1 — X2 and Y1 m YE; based on which you want to estimate the ratio of standard deviations 01/02. (1) Show that M H(X1,X2:Y1a33="1/02) I er WY2 |01 is a pivot. What is its distribution? (ii) If 01 2 02 = (I, calculate the distribution of (X1 - X2)2 + (Y1 — Yzlg 2 0'2 and indicate how you can construct a 0.1. for 02 based on the above expression. (5 + 5 = 10 points) Let X1,X2, . . .,Xn be i.i.d. Uniform(—191 6), where 6 > 0. Find a MOM and the MLE of 19. Is the MLE unbiased? (10 points) Consider i.i.d. observations X1, X2,. . . ,Xn where each X1- follows a normal distribution with mean and variance both equal to 1/9, where 6 > 0. Thus, Show that the MLE is one of the solutions to the equation: = 82Ww6e1=0 Where W 2 71—1 217:1 XE. Determine which root it is and compute its approximate variance in large samples. ...
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final--exam - IS my FINAL EXAM: STATISTICS 426 Moulinath...

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