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' Probability OLD (i/ (/{ 6. For each of the following experiments and events determine the probability of the events. : (a) (4 pts.) Experiment: Four 6-sided fair dice are thrown Event: All four numbers appearing are different (b) (6 pts.) Experiment: Three balls are chosen at random without replacement from an umn containing 20 red and 40 white balls. Event A: Exactly one red ball in the sample EventB: 3%ballisred Are A and B independent? Support your answer with analysis. .(c)(4pts.) -Experiment: Balls are drawn at random without replacement from the urn described in (b) until 2 red ball is selected. Event: A red ball is first selected on draw number 7. (d) (6pts.) Experiment: X and Y are independent, each uniformly distributed on the unit interval [0, 1] Event : X >YIN[X+Y >n/4} (e) (6 pts.) Experiment: Four observations are taken independently from the normal distribution that has mean 50 and standard deviation 10. ’ Event A: Al four observations exceed 35. Event B: The first observation is less than the average of the last three observations.

© 7.(8 pts.) Let X, Y have the following joint probability mass function: .~ 'Obtain the following: (a) The marginal probability mass functions of X and of Y (b) PIX > 0) v (Y > 0] (©) P[Yisodd ‘ X is even] (d) EXY) _ 8. Let X have the density f(x)= 1 forO0<x <1 = 0 otherwise ﬂa\)?(4 pts.) Find the c.d. £. F(y) for Y = 5X - 1 R R (b)’ (4 pts.) Find Var(Y). (;) (6 pts.) Find-a function h suchithat Y = h(X) has density g(y) = » 3y for0 <'”5 <1 = 0 otherwise

9. (6 pts.) Let X, have a uniform distribution on the interval {0. I/m) forn=1.2. ... Prove that {X,} converges in probability to 0. 10. :Consider the experiment where X distributed Poisson (X) is observed and. conditional on X = X, a coin with probability = of Heads is given X independent ‘tosses. Let Y denote the number of Heads among these tosses. In solving (a) and (b). - you may use the fact that a Poisson (1) has mean A and variance A and a Binomial (n, p) has mean np and variance np(1-p). You may choose to{c) first and deduce (a) and (b) after having established (c). , (’.(;)f)(z pts.) Find E(Y). . @(4 pts) Find V(Y). ( (c) )(6 pts.) Show that Y is distributed Poisson (A7). Statist_ics 11) (6 pts) Let X;. X, ... , X, be arandom sample from a distribution with mean p. variance n " 0% LetS’= Z(x; - X)) /(n - 1). Prove that E( $%) = ™. i=1 12) Let X;, Xa, ... , X, be a random sample from thé distribution with density f(x;0)=0 x® 1 for0<x<1, where 9> 1. o 5) (5 pts.) Find the method of moments estimator of 0. b) (6 pts.) Find the maximum likelihood estimator of 6. ( c) 38_pts‘.) Use the fact (you need not pfove it) that Y = - log(X,) has the exponential ) . “distribution with mean 1/6 to find an approximate 95% confidence interval on 0 for n = 100. . Hint: The exponential distribution with mean p has variance pt°. @)\(8 pts.) Let Hp: 0 =2 and H,: 6 =3. Give the a0 = 0.05 level most powerful test of Ho vs H,. J i . '\/c) @pts)Is this test uniformly most powerful for Hp: 8 =2 vs Ha: 9 > 27 Why? 3

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13) X have the binomial distribution with parameters n and p. with n known, p unknown. O<p<li. (a) (6pts.) Letp p = X/n. Find an approximation for the smallest n for which P(]S, - p]é’ 0.02_) >.90. ( b) (5 pts ) Suppose you wish to test of Hy: p <0.30 vs H p>0. 30. Give an approxxmate o = 0.05 level test ofHo vsH, forn= 100. (10% 6] pts.) Let @ be the cdf for the standard normal dlsmbution. Express the power function P) of the test in b) in terms of 11> . 14) A statistics professor gave his class an assignment: Drop two thumb tacks on a hard surface, and count the number of the tacks for which the point is upward. Repeat this experimcnt 1000 times. a) (8 pts.) The professor suspccted that student XYZ, being lazy, would just make up the sults, so he decided to perform an = 0.05 level goodness-of-fit chisquare test. XYZ's results were: - X =0 occurred 80times, X =1 440times, X=2 480 times State a null hypothesis and then carry out the test. Hint: Under the null hypothesis the MLE for “ the probability that the tack point is upward is the proportion of occurrences of “up” for all tacks thrown. ’ " b) ¢ pts ) The professor susp&cied that othefs in his class of 30 had cheated, so he performed the { for each student. He rejected the null hypothesis for two of the 30 students and therefore failed those two students. If you were the lawyer for those two students, what argument would you _use to convince a jury that failing them was unfair?

15) In order to determine whether a three week course designed to improve students’ scores on the verbal part of the graduate record exam (GRE) was effective 20 students were chosen randomly. These students were then given a one hour verbal test, then paired according to their scores, the two with the highest scores being in pair one, those with the next highest in pair 2, and so on. Then one member of each pair was chose randomly using a coin flip and placed in the treatment group. The others were " placed in the control group. The 10 students in the treatment group were given the three week course. The others did not take a course. After the course ended all 20 students took:the GRE verbal test. Their scores were 10 times the follows. Use these numbers for any analysis. Pair 1 2 3 4 5 . 6 7 8 9 10 Treatment 73 67 65 60 61 54 53 45 46 43 64 | 66 | 57 56 54 52 44 47 39 { Control 71 . @(10 pts.) State a mbaei then test Hy: no treatment effect vs H, treatment improves scores for o = 0.05. - /t:)}S pts.) Give a 95% confidence interval on the mean treatment effect 8. 16) Suppose that (Xj;, X2;, Y;) are observed fori= 1, 2, ..., 6. where the x;; and Xin are constants and are constants and Y; is a random variable, where “Y; =By xy5 + B2 X2 + &. Suppose that the &; are independent. each with the N(0, 02) distribution. The observations are: i 1 2 3 4 5 6 Xii 0 0 1 1 2 2 Xn 1 -1 0 0 -1 1 Y; O 0 2 2 6 2 @ /(8 pts.) Find'the least squarés estimates of 8, and f3,. ’ e A . 6)/.(5 pts.) Find the residuals e; = Yi - Y;, and use these to estimate o2 { / —'c) (5 pts.) Find a 95% confidence interval on B2. 7

10. 11. 12. {3. Probability (5 pts) Show that if P(E;) =1 forall i =1, 2, ...,then P(n2, E)=1. (5 pts) You roll a pair of fair dice. One of them is red and the other is green. Consider the following three events: . A = {an even number shows on red die}, B = {an odd number shows on green die}, . C = {the product of the two dice is even}. Are these three events independent? You must give the right reason. (6 pts) Consider independent Bernoulli trials, each resulting in a success with probability p; are performed. What is the probability of » successes occurring before m failures? (12 pts) Find the density function of X + Y where (X,Y) has the joint density f(X,Y)=%J’e_‘7 0<x<ew, 0<y<2. (10 pts) Let’X and Y be independent normal N(ut, o) random variables. Show that X + Y is independent of X-Y. (12 pts) Let X and Y be two independent exponential random variables with parameter A > 0. Find the density function of X + Y. : (5 pts) Urni A has 6 white and 4 black balls. Urn B has 5 white'and 12 black balls. We flip a fair coin. If the outcome is heads, then a bali from umn A is randomly selected, whereas if the outcome is tails, then a ball from urn B is selecied. Suppose a black ball is selected. What is the probability that the coin landed tails? (10 pts) The lifetimes of interactive computer chips produced by a certain company are normally distributed with parameters jt = 1.4 x 10° hours and 6% = 9 x 10'® hours. What is the approximate probability that a * batch of 100 chips will contain at least 20 whose lifetimes are less than 1.8 x'10° hours?

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Statistics 14. (15 pts) Suppose that X ~ Exponential () and, conditional on X = x, ¥ ~ Exponential (6 ), that is, fo(0)=68"% x>0 and fy(y|x)=6e®,y>0 where @ > 0. Let (X; Y),i=1,2, ..., n be iid with the same distribution as (X, ). (a) Show that 7 = J-Z"X ;(1+Y;) is sufficient for 6. 2n t . (b) Find the mean and variance of 7. Recall that the Exponential (#) has mean 1/ and variance 1/6%. 15. (15pts) Let Xand ¥, =1, 2, ..., i be independent Bernoulli random variables with X; ~ B(1, z) and Y~ B(1, l-m)with0< 71, ’ (a) Show that 7= 3", - 3", is sufficient for . (b) Determine the mle of n'based_ on Z, and its mean and variance. : '(c) Is the mle consistent estimator of 7 in this case? -Explain your answer. 16. (15 pts) Suppose thatX i=1,2,..., nare iid Poisson (1). You will have to use the Poisson tables for Part (a) the normal tables for Part (b). (a) With# = 5, determine the rejection reglon of the level .005 UMP test of Ho =1vs. H;: 2> 1 and the . power of the testat 1= 1.4, . . (b) With = 25, determine the approximate rejection region of the level .005 UMP test-of Hy: 1=1 vs. H;: 2> 1.and the power ofithe test at 1= 1.4. 17. (20 pts) Consider the linear regression fnodel Yi=fxi+e, i =1,2,., nwhere the x; are constants and the _random errors e; are independent with E(e)) = 0, V(e) = o ‘ ~(a) Derive the least squares estimate of the slope 4. (b) Derive the mean and the variance of the estimate. (c) Determine the 95% confidence interval estimate of £ under the usual normal theory assumption on the random errors and evaluate itor the n= 6 data: 1,1, 2), (2,3),(2,5),(,6),(3, 7). 7

1078 / Appendix tables TABLE A-4 (continued) Percentiles of the F distribution Numerator df Den df A 1 2 3 4 5 6 7 8 9 1 .50 1.00 150 LTI 1.82 1.89 1.94 1.98 2,00 2.03 .90 39.9 49.5 53.6 55.8 5712 58.2 58.9 59.4 59.9 95 161 200 216 225 230 234 237 239 T 4 975 648 800 864 900 922 937 - 948 957 963 99 4,052 5000 5403 5,625 © 5764 5859 5,928 5981 6,022 995 | 16,211 20,000 21,615 22,500 23,056 23,437 23,715 23,925 24,091 -999 |- 405,280 500,000 540,380 562,500 576,400 585,940 392,870 598,140 602,280 2 .50 | . 0.667 1.00 1.13 1.21 1.25 1.28 1.30 132 133 .90 8.53 9.00 9.16 9.24 9.29 9.33 9.35 9.37 9.38 i 95 18.5 19.0 19.2 19.2 19.3 193 19.4 194 19.4 ‘ 975 38.5 39.0 39.2 39.2 39.3 39.3 394 39.4 394 .99 98.5 99.0 99.2 99.2 99.3 99.3 99.4 99.4 99.4 995 | - 199 199 199 © "199 199 199 199 199 199 ! 999 998.5 9990 9992 9992 9993 999.3 9994 9994 999.4 3 .50 0.585 0.881 1.00 1.06 1.10 1.13 1.15 1.16 1.17 ! .90 5.54 5.46 5.39 5.34 5.31 5.28 5.27 5.25 5.24 .95 10.1 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 975 17.4 16.0 154 15.1 14.9 14.7 14.6 14.5 14.5 99 34.1 30.8 29.5 28.7 28.2 27.9 27.7 275 273 995 55.6 49.8 41.5 46.2 45.4 44.8 444 44.1 43,9 999 1670 1485 1411 137, 1346 1328 1316 1306 129.9 4 .50 0549 0.828 0.941 1.00 1.04 1.06 1.08 1.09 1.10 .90 4.54 4.32 4.19 4.11 4.05 4.01 3.98 3.95 3.94 95 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 975 122 10.6 9.98 9.60 9.36 9.20 9.07 8.98 8.90 .99 21.2 18.0 16.7 16.0 15.5 15.2 15.0 14.8 14.7 995 313 26.3 243 23.2 22.5 220 21.6 214 21.1 999 74.1 61.2 56.2 53.4 51.7 50.5 49.7 49.0 48.5 5 .50 0.528 0.799 0.907 0.965 1.00 1.02 1.04 1.05 - 1.06 .90 4.06 3.78 3.62 3.52 345 3.40 3.37 3.34 3.32 A 95 | 661 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 ' 975 10.0 8.43 7.76 7.39 7.15 6.98 6.85 6.76 6.68 .99 16.3 13.3 12.1 11.4 11.0 10.7 10.5 10.3 10.2 995 22.8 18.3 16.5 15.6 149 . 145 14.2 14.0 13.8 .999 47.2 371 332 311 29.8 - 28.8 © 282 276 272 6 .50 0.515 0.780 0.88¢ 0.942 0.977 1.00 1.02 1.03 1.04 .90 3.78 3.46 3.29 3.18 3 3.05 3.0t 2.98 2.96 95 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 975 8.81 7.26 6.60 6.23 5.99 5.82 5.70 5.60 5.52 .99 13.7 10.9 9.78 9.15 8.75 8.47 8.26 8.10 7.98 995 18.6 14.5 12.9 12.0 11.5 11.1 10.8 10.6 10.4 999 355 27.0 23.7 21.9 20.8 20.0. 19.5 19.0 18.7 7 .50 0506 -0.767 0.871 0.926 0.960 0.983 1.00 1.01 1.02 90 3.59 3.26 3.07 2.96 2.88 2.83 278 2.75 272 95 5.59 4.74 4.35 4.12 3.97 3.87 3.79 373 3.68 975 807 - 6.54 5.89 5.52 5.29 5.12 4.99 4.90 4.82 .99 12.2 9.55 8.45 7.85 7.46 7.19 6.99 6.84 6.72 995 16.2 12.4 10.9 10.1 9.52 9.16 8.89 8.68 8.51 999 29.2 21.7 18.8 17.2 16.2 15.5 l_.5.0 14.6 14.3

STT 315 Table of Cumulative Poisson : vcm\mdmo.og Sections 1 - 22 . - | : mu Afi : Xx}101/02/03|04{05/06]07/08109] 1 [11{42{13]14[15]16[17[18{19, 2 121]22!123/24,25]|26!27 0,5048| 0.8187( 0,7408) 0,6703| 0.6065| 0,5488| 0.4966 0.4493)0.4066| 0.3879) 0.33290.3012} 0.2725| 0.2466 0.2231| 0.2019 a;qu 0.1653] 0.14961 0.1353] 0,1225|0.1108} 0.1003} 0.0907|.0.0821] 0.0743} 0.0672 0.995310.9825| 0.9631 0.9384/ 0.9098| 0.8781} 0.8442 o.momw 07725 o.\umc 0.6990(0.6626 0.6268.0.5918| 0.5578 0.5249 0.4932) 0.4628 o.nw.mq c..aomo o.wqwm oumam 0.3309 0.3084 c..mmqm 0.2874|0.2487 0.9998| 0.9988( 0,9964 | 0,9921| 0.9856| 0,8769 o.mmmw menm o.wuq_ 0.9197 o.woom owqom 0.8571(0.8335 0.8088(0.7834{ 0.7572| 0.7306 oqomw o.mumﬂ 0.6496| 0.6227}0.5960 0.5697| 0.5438| 0.5184] 0.4936 1.0000 0.99899}0.9997 0.9992/ 0,8982| 0,9986{0.9942) 0.9909 vobmmw 0.98101 0.9743 c.ommn 0,9569(0,9463| 0.83440.9212}0,9068]0.8913 obqfi omm.wé o.m.umw. mm_ma 0.7993|0,7787{0.7576 0.7360] 0.7141 1,0000/ 1.0000 %.0090} 1,0000 0.8998|0.6898{ 0,9952| 0.9986| 09877 0.9963 0.9946| 0,9923) 0.9893| 0,9857} 0.8814| 0.9763| 0.9704;0.9636 | 0.9558 0.94730.937910.9275) 0.9162| 0.9041 o.mm.m. o.mq.q.u 0.8629 1.0000( 1.0000} 0.9999 1,0000 1.0000/ 1.00001 1.0000 ._m.ooc 1.0000 1,0000 1.0000| 1.0000] 1,0000; 1.0000 ‘oooc 1.0000| 1.00004 1.0000( 1,0000| 0.9999| 0.9999{ 0.9998| 0.9897 | 0.9996 0.9994| o”wmwu 0.9980 QN RN DIWIN. 2. D 0.9994] 0.9990| 0.9985| 0.9978| 0,9968) 0,9955| 0.9940| 0,9920} 0.9396 0986809834} 0.9796/0.9751 c.w.qoo 0.984310.9580} 0.9510}0,9433 09999 0.9909| 0,997 | 0.9996] 09994 0.9891 | 0,9967 | 0.9981 | 0.0674) 0.9966 0.9794 1.0000 0.8934 1,0000] 1.0000] 1.0000) 1.0000 1.0090) 100001 1.0000] 1.0000] 1,090 +.0000) 1.0000] 1.0000] 1.6000] 10000 1.0800] 1.0000| 0.8595) 0.98%¢ | 00884 0,998 0.9995 0.9984 ""8|1.0000| 1.0000| 1.0000| 1.0000] +.0000] 1.0000] 1.0000| 1.0000| 1.0000| 1.0000] 1.0000] 1.0000 1.0600} 1.0000] 1.0000 | 1.0003| 1.0000] 1.0560| 1,0000| 4,0000| 0.8888 0,688 0.9995 10 1.0000] 10000 1.0000] 1.0000 1.0000] 1.0000 1.0000] 1.0000] 1.0000| 1.0000] 1.0000] 1.0000] 1.0000] 1.0000] 1.0000] .0000| 1.0000] 1,000 1.0000] 1.0000 | 1.0000] 10000 .0000| 1,000/ 0.8599 | 09968 0,099 1.0] 10000 1 1.0090{ 1.0000| 1.0000{ 1.0000] 1.0000/ 1 0000 1.8600] 7.0000] 1.0000] 1.5000] +.0000 1.6000 1,000 0008 10008 1.0000] 1.0080] 1.0030] 1.0000 1,000 1.0600 00 12| 1.0000 1.0000 1.0000 1 0000 I 10 1,000 1.0000 1.0000| 1.0000 v " 13 1.0000] 1.0000} 1.0000 1.0000| 1.0000{ 1.0000 1.0000{ 1.0000] 1.0000 000/ 1.000¢ 1.0000} 1.0000}1.0000 1,0000{ 1.0000] 1.0000] 1 0000 00} 1.0000 1.0000) 1.0000| 1.0000 1.0000 1.0000 1.0000} 1.0000| 1.0000¢ 1.0000 A.ecoa *.oooo _.Qooo 1.0000 ‘oooo 1.0000{ 1.0000| 1.0000| 1,0000 1.0000| 1.0000| 1.0000/ 1.0000] 1.0000; 1.0000| 1.0000 1.0000{ 4.0000| 1.0000| 1.0000| 1.0000} 1.0000| 1.0000 1.0000| 1.0000] 10000/ 1.0000{ 1.0000| 1.0000| 1.0000} 10000 1.0000| 1.0000| 1.0000{ 1.0000{ 1.0600 "+ 14]1.0000] 1.0000] 1.0000 7 15| +.00001.0000 1.0000 16| 1.0000| 1.0000 1.0000 4711.0000| 1.0000] 1.0000 ¢ , .. 1.0000 1914 308 1.0000{ 1.0000 20/ 1.0000{ 1:0000] 1.0000 N‘:,.SS 10000/ 1.0000 N,M_A_.od.cc 1,0000 __.c.coc 231,000 1.0000| 1,0000 1.0000| 1.0000 1.0000| 1.0000|1.0000| 1.0000{ 1.0000| 1.0000| 1.0000 1.0000| 1.00001 1.0000/ 1.0000] 1.0000 1.0000: 1.0000 1.0000/ 1.0000| 1,0000| 1.0000} 1.0000| 1.0000| 1.0000 1.0000 1.00001 1.0000 ._.coom 1.0000 10000 1.0000( 1.0000{ 1.0000 1.0000 1.0000 1.0000( 1.0000 ._oooo 1,0000 4080 ..oooo 1.0000( 1.0000| 1.0000 1.6000{ 1.0000 1.0000{ 1.0000 1.0000| 1.0000| 1.0000 __oooo ._oooo Aooao 1.0000| 1.00001 1.0000 d,coom 1.0000; 1.0000 1.0000] 1.0000 1.0000| 1.0000{ 1.0000| 1.0000{ 1.0000| 1.0000{ 1.0000; 1.0000. 1.60004 1.0000 Aoooo ._oooo 1.0000¢ 1.0000| 1.0000 1.00004 1.0000 aoooo 1.0000| 1.0000| 1,0000 1.0000; 1.0000, 1.0000/ 1.0000} 1,0000{ 1.0000 1.0000] 1.0000] 1.0000, 1.0000 aoooc 1.0000 #oooo 1.0000 ._oooc 1.0000 ‘_oaoo 1.0000/ 1,0000 1.0000| 1.0000{ 1.0000| 1.0000 ﬂoooo 1.0000} 1.0000 d.o.coo 1.0000{ 1.0000| 1.0000 aoeoc dooco 1.0000 aoooo 1.0000} 1.0000| 1.0000 1.0000} 1.0000} 1.0000| 1.0000 4oooo 10000/ 1.0000| 1.0000/ 1.0000} 1.0000 1.0000! 4 ‘oooo 1.0000 1.0000/ 1.0000| 1.0000| 4.0000} 1.0000 Aocoe“__ oooo 10000 1.00004 1.0000 1,0000 1.0000| 1.0000( 1.0000} 1.0000| 1.0000; 1.0000 _booo 1.0000} 1.0000( 1.00001 1.0000 #oooo;,cooo 1.0000; 1.0000| 1.0000{ 1,0000| 1.0000( 1.0000 ‘_oooo 1.00001 1.0000} 1.0000| 1,0000, 1 0000 1 0000 1.0000| 1,0000 ,_.oooo 1.0000( 1.0000 ?ocoo 1.0000} 1.0000 1.00001 1.0000 _cooo #o@oc 1.0000( 1.0000{ 1.0000| 1.0000 1.0000| 1.0000] 1,0000| 1.0000] 1.0000| 1,0000 acooc_ 1.0000] t 0000 1.0000 A.ooco_ 10000 10000 251 1.0000} 1.0000} 1.0000] 1.0000} 1,0000] 1.0000] 1.0000{ 1,0000{ 1.0008] 1.0000| 1.0000| 1.0000] 1.0060| 1.0000] 1.0000| 1.0000{ 1.0000 1,0000{ 1.0000 Aoooo:oooo 1.00001 1.0000; 1.0000 1.0000 $.0000 1.0000 | B .&.%u@\.seip N= L7, PiX23)= ,q0¢%

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S8TT 315 , Sections 1~ 22 57 | 58 | 59 6 61|62 ;63|64;65] 66| 67 0.0033 0.0025 0.0022] 0.0020] 0.0018] 0.0017] 0.0015| 0.0014 00012 0.0224 “o0174] 0.0150| 00t4s| oo134| 00123 00113]- o0103] 0.0085 ) 0.0820| 0.0577| 0.0536| 0.0498] 0.0463| 0.0430] 0,0400| 0.0371 0.1612| 0.1425| 0.1342| 0.1284] 0.1189] 0.1118| 0.1052| 0,088 0.3422 0.2851] 0.2718| 0.2582| 02468 02351} 0.2237| 0.2127| 0.2022 0.5289} 0.5(19| 0.4850 04457| 04298| 04141| 0.3088| 03837 0.3650| 0.3547| 0.3408 0.6660| 0.6763| 06544 0. 6.6063| 0.5902] 05742] 0.6882| 0.5423| 0.5285| 0.5108] 0.4953 0.8095| 0.7870| 0.7841| 0.7710| 0.7578| 0.7440] 0.7301|, 0.7180] 0.7017| 0.6673| .6728| -0.8581] 0.6433 0.8944 0.9867| 0.9766| 0.8672| 0.8674] 0.8472| 0.8267) 0.8256] 0.8148| 0.8033| 0.7918| 0.7798 0.7673 .9452| 0,0409| 0.9352| 0.9202| 0.9228) 0.9161] 0.9090| 0.9016| 0,8808| 0.9858| 0.8774| 0.8688| 0.8598| 0.8502 0.9718| 0.0685| 0.9551] 0.9614| 0.9574| 09531 0.0486] 0.0437| 0.9386] v.easz| 0.e2ra| v.e214| V15t 0| 0.9875| 09855 0.9841] 0.9821] 0.9700| a.e776] v.o750] 0.0720| 0.0683| 098e1] 0.9627| 0.05e1| nessy| e e s oo, e s s3] 0580 09877 ssy7| osezs] 0se20| 08| ossss| 0cess 08983/ 08691 0.9074) 0.8970] 0.9986| v.9ve1| 0.985e| 0.0950 0.9988| 0.9907 0.9860 5| 0.9984| 0 0 0.9999| 09999 [ 0.9094| 09093 .0000| 1.0000| 1.0000] 1. 0.8999 0.ooee| 0.098] 0.0998! 0.9957| 10000} 1.0000 1.0000] 1.0000| 1.0000| 1.0000| 1.0000| 1.0000| 1.0000) 0.99%8] 0.9908| 0.9905| 0.0998 1.0000] 1.0000] 1. 1.0000{ 1.0000] 1.0000| 1.0000] 1.0000] 1.0000| 1.0000| 1.0000| 10000 1.0000| 1.0000| 4.0000 "1.0000| 1.0000] 1,0000| 1.0000| 1.0000| 1.0000 1.0000] 1.0000| 1.0000| 1.0000| 1.0000, ;:0000 1— 1.0000(. 1.0000| 1.0000{ 1.0000 1.0000-1.0000 “1.0000 1.0000{ 1.0000: ’ 0 1.0000| 1.0000| 1.0000] 1.0000|. 1.0800| 1.0000| 1,000 1.0000| 1.0000| 1.0000| 1.0000| 1.0000| 1.0000 10000 1.0000 1.0000| 4.0008] 1.0000| 1.0000] 1.0000| 1.0000] 1.0000| 1.0000| 1.0000] 1.0000| 1.0000| 1.0000| 1.0000] 1.0000 1,0000] 1.0000] 1.0000| 10000} 1.0000| 1.0000| 1.6000| 1,0000| 1.0000] 1.0000] 1.0000{ 1.2000| 1.0000| 1.0000|. 1.0000] 1.0000 1.0000| 1.0000] 1.0000] 1.0000| 1.0000| 1.0000] 1,000 1.0000] 1.0000] 1.0000] 1.0000| 1.0000| 1.0000 1.0000| 1.0000 1.0000 02/27/2001 //

Probability 4. For each of the following experiments and events determine the probability of the event. Assume all (the dice are equally likely to result in any of the numbers 1,2,..,6. (2) (4 pts.) Experiment: Five 6-sided dice are thrown Event: At least two 6’s occur. (b) (7 pts.) Experiment: 600 6-sided dice are thrown Event : - At Jeast 120 of the tosses result in 6. Use an approximation. (c) (7 pts.) . Experiment: 600 -6-sided dice are thrown. Event: - The total of all the numbers obtained is less than 2000. Use an approximation. Hint. The variance of the uniform distributionon 1,2, ... |6 is 35/12. (d) (4 pts.)) Experiment: Two dice are thrown until 6-6 occurs. . Event: The first 6-6 occurs on throw number 5. (e)(7pts.) Experiment: Two dice are thrown 100 times. - Event : Fewer than two 6-6°s occur. Use an approximation. " (H(3pts.) Experiment: A box has 10 eggs. 3 of these are rotten. 4 eggs are drawn randomly without replacement. Event: The sample contains 3 rotten eggs S. (8 pts.) A test for the HIV virus has probability 0.99 of being positive if the person really has the virus. It - has probability 0.005 of being positive if the person does not have the virus. Among all people taking the ‘test 0.02% (1/50 of 1%) actually have the virus. If one of these people is chosen randomly and the test is positive, what is the conditional probability that the person has the virus? : x for0<x <1 6. Let X have the density f(lx)= % forl<x <2 0 otherwise’ () (5 pts) Find the-c.d.f. F(x) for X. (b) (5 pts.) Find Var(X). (¢) (5 pts.) Find the density of Y = X2 . 7. Let X, have cdf F, forn=1,2, ... (2) (5 pts.) Define: F, converges in distribution to a cdf F. (b) (5 pts.) Let X, take the values 1/n and n with probabilities 1 ~ 1/n and 1/n. Does {X,} have a limiting distribution? If so, what is it? Show it. - (c) (5 pts.) Let Y,, be uniformly distributed on 1/n, 2/n, ... > 0/n. Does Yy, have a limiting distribution? If s0, what is it? Show it. /2

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Statistics 8. LetX; X, Xn tea random sample from a normal distribution N(y, o® = 100). ‘(The normal distribution has variance 100.) (2) (5 pts.) Find the rejection region of the uniformly most powerful o = .05 test of Ho: p=0vs. Hy: p>0. M) ¢ pts.). What is the power of the test at p =2 if n =257 (©) (5 pts.) What must n be so that the power at 1 =2is 807 9. LetZ=X+Y where X ~B(m, .8) and Y < B(200 - m, .2) are independent binomial random variables and m is unknown, m € {0, 1, ..., 200}. (a) (4 pts.) Use the method of moments to find an estimator of m based on Z. (b) (4 pts.) What are the expected value and the standard deviation of your estimate? “(¢) (4 pts.) Construct an approximate 95% CI estimate of m assuming the realization z = 124. 10. Let X;, Xo, ..., X, be a random sample from ™) - f(x16) =le"’”, x>0 - where 6> 0. (@ (4 pts.) Determine the expectation E, o(X) and the standard deviation SDOO() of the distribution with - * density (*). ¢ p_ts.) Find 6 the mle of @ based on X, Xs, ..., X MLE ? N (c) (6 pts.) For large n, determine an approximate 99% CI on 0 based on Ovie /3

11. Consider the sample space of four points and the two (discrete) distributions given below. Consider testing the simple versus simple hypotheses Ho: X ~ Pg vs. Hy: X ~ Py The Neyman Pearson tests have Rejectlon Regions of the form {x f’P A(x) <c} for some ¢ where A(x) = po(X)/p1(X). X - Xy X2 X3 X4 po(x) 0.1 0.1 0.7 pi1(x) 0.2 0.4 0.1 Ax) (8) (2 pts.) Compute the likelihood ratio A(x) for each x and place the values in the table above. (b) (6 pts.) Of all possible tests of size o = .2, which is most powerful? (c) (2 pts.) What is the power of the most powerful test of size o = .27 (d) (2 pts.) Find a test-of the same size but with less power than that of the NP test. 12. Consider the standard normal theory regression model with regressmn function E(Y{x)=Po+ B1x+ Py x* and the data x 1 N 0 0 1 1 v 3 9 15 13 8 12 (a) (6 pts.) Find the least squares estimates of the parameters Bg, Bi, and p. (b) (4 pts.) Determine SSE. A (c) (4 pts.) Find the estimated variance of B, (d) (4 pts.) Give a 95% CI estimate of B,. (&) (4 pts.) Carry-out the .05 level F-test of Hy. By =, =0. /4

1074 |/ Appendix tables TABLE A-2 Percentiles of the t distribution A P Eatry is i(A; v) where P{t(v)=<t(A;v)} = A A v 60 70 80 .85 90 95 975 1 0.325 0.727 1.376 1.963 3.078 6.314 12.706 2 0.289 0.617 1.061 . 1.386 1.886 2.920 4.303 3 0.277 . 0.584 0.978 1.250 1.638 2.353 3.182 4 0.271 0.569 0.941 1.190 1.533 2.132 2.776 5 0.267 0.559 0.920 1.156 1.476 2.015 2.571 6 0.265 0.553 0.906 1.134 1.440 1.943 2.447 7 0.263 0.549 0.896 1.119 1.415 1.895 2.365 8 0.262 0.546 0.889 1.108 1.397 1.860 2.306 9 0.261 0.543 0.883 1.100 1.383 1.833 2.262 10 0.260 0.542 0.879 1.093 1.372 1.812 2.228 11 0.260 0.540 0.876 ° 1.088 1.363 1.796 2.201 12 0.259 . 0.539 0.873 1.083 1.356 1.782 2.179 13 0.259 0.537 0.870 1.079 1.350 1M 2.160 14 0.258 0.537 0.868 1.076 1.345 1.761 2.145 15 0.258 0.536 0.866 1.074 1.341 1753 . 2.131 16 0.258 0.535 0.865 1.071 1.337 1.746 2.120 17 0.257 ° 0.534 0.863 1.069 1.333 1.740 2.110 8 0.257 0.534 0.862 1.067 1.330 1.734 2.101 19 0.257 0.533. 0.861 1.066 1.328 1.729 2.093 20 0.257 0.533 0.860 1.064 1.325 1.725 2.086 21 0.257 0.532 0.859 1.063 1.323 1.721 2.080 22 9.256 0.532 0.858 1.061 1.321 1.717 2.074 23 0.256 0.532 0.858 1.060 1.319 1.714 2.069 24 0.256 0.531 "0.857 1.059 1318 L1 2.064 25 0.256 0.531 0.856 1.058 1.316 1.708 2.060 26 | 0.256 0.531 0.856 1.058 1.315 1.706 2.056 27 0.256 0.531 0.855 1.057 1.314 1.703 2.052 28 0.256 0.530 0.855 1.056 1.313 1.701 2.048 29 0.256 0.530 0.854 1.055 1.311 1.699 2.045 30 0.256 0.530 0.854 1.055 1.310 1.697 2.042 40 0.255 - 0.529 0.851 1.050 1.303 1.684 2.021 60 0.254 0.527 0.848 1.045 . 1.296 1.671 2.000 120 0.254 0.526 0.845 1.041 1.289 1.658 1.980 © 0.253 0.524 0.842 1.036 1.282 1.645 1.960

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9. (6 pts.) Let X, have a uniform distribution on the interval [0. I/n) forn = 1.2, ... Prove that { X} convergesin probability to 0. 10. -Consider the experiment where X distributed Poisson (A) is observed and. conditional on X = X, a coin with probability 7 of Heads is given X independent “tosses. Let Y denote the number of Heads among these tosses. In solving (a) and (b). vou may use the fact that a Poisson (4) has mean A and variance A and a Binomial (n, p) has mean-np and variance np(1.- p). You may choose 10(c) first and deduce (a) and (b) after having established (c). (@ /2 pts.) Find ECY). @(4 pts.) Find V(Y). ((© X6 pis.) Show that Y is distributed Poisson (A Statistics 7 11) (6 pts) Let X; X», ... , X, be arandom sample from a distribution with mean p. variance n o’ Let§* = Z(xi - XY /(n - 1). Prove that E( §%) = ¢™. i=1 12) Let X4, Xa, ... , X, be a random sample from the distribution with density f(x;0)=0 x® ! forO<x<1,where 8> 1. ‘a) (5 pts.) Find the method of moments estimator of 6. b) (6 pts.) Find the maximum likelihood estimator of 6. ( c) (8 pts.) Use the fact (you need not prove it) that Y = - log(X;) has the exponential . “distribution with mean 1/8 to find an approximate 95% confidence inteérval on 6 for n = 100. . Hint: The exponential distribution with mean p has variance p:. Cfis pts) LetHo: 0 =2 and H,: 6 =3. Give the o= 0.05 Jevel most powerful test of Ho vs Ha. J / Ve)fi (2 pts.) Is this test uniformly most powerful for Ho: 0 =2 vs H,: > 27 Why? /o

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