1. True or False? (20 pts) For parts (a)-(j), write a “T” or “F” under the statement. (a) (2 pts) Consider a Poisson process with rate λ, where N (t1 , t2 ] is the number of events between t1 and t2 . Then if N (0, 1] → ∞, N (1, 2] → 0. (b) (2 pts) If X is a random variable, E[X] is its expectation, and h(X) is some function of X, it is always true that E[h(X)] 6= h(E[X]). (c) (2 pts) Assuming they exist, knowing any one of the mgf, cdf, or pdf means you can find the other two. (d) (2 pts) If X and Y are two normally distributed random variables, Z = X + Y is normally distributed. t2 (e) (2 pts) The mgf of a standard normal random variable is e 2 . (f) (2 pts) The mode of the Chi-squared distribution is always greater than zero. (g) (2 pts) In order to prove Fisher’s Approximation Theorem, you need both the Central Limit Theorem and the Law of Large Numbers. (h) (2 pts) The two limits on a confidence interval, L̂ and Û , contain the same information as a P-value. (i) (2 pts) If [L̂, Û ] constitute a 95% confidence interval for θ̂, then the probability that [L̂, Û ] contains the state of nature θ is 0.95. (j) (2 pts) The χ2 statistic can always be used for hypothesis testing on the multinomial distribution. 2 2. An MLE Problem. (30 pts) Let X1 , ..., Xn be i.i.d sample from a Uniform[0, θ] distribution. (a) (4 pts) Find θ̂, the MLE of θ. (b) (12 pts) Find the density function of θ̂ in (a). (c) (14 pts) Is the MLE unbiased? Find the bias, variance and mean squared error of the MLE. 5 4. Testing a Hypothesis (30 pts). Let X1 , X2 , ..., Xn be an i.i.d sample from an exponential distribution with parameter θ such that f (x|θ) = θe−θx . (a) (15 pts) Derive a likelihood ratio test for H0 : θ = θ0 versus H1 : θ 6= θ0 . (b) (15 pts) Show the rejection region is of the form X̄e−θ0 X̄ ≤ c, where X̄ is the sample mean. 11 Φ(zp) zp Cumulative Normal Distribution z 0.0 0.1 0.2 0.3 0.4 .00 .5000 .5398 .5793 .6179 .6554 .01 .5040 .5438 .5832 .6217 .6591 .02 .5080 .5478 .5871 .6255 .6628 .03 .5120 .5517 .5910 .6293 .6664 .04 .5160 .5557 .5948 .6331 .6700 .05 .5199 .5596 .5987 .6368 .6736 .06 .5239 .5636 .6026 .6406 .6772 .07 .5279 .5675 .6064 .6443 .6808 .08 .5319 .5714 .6103 .6480 .6844 .09 .5359 .5753 .6141 .6517 .6879 0.5 0.6 0.7 0.8 0.9 .6915 .7257 .7580 .7881 .8159 .6950 .7291 .7611 .7910 .8186 .6985 .7324 .7642 .7939 .8212 .7019 .7357 .7673 .7967 .8238 .7054 .7389 .7704 .7995 .8264 .7088 .7422 .7734 .8023 .8289 .7123 .7454 .7764 .8051 .8315 .7157 .7486 .7794 .8078 .8340 .7190 .7517 .7823 .8106 .8365 .7224 .7549 .7852 .8133 .8389 1.0 1.1 1.2 1.3 1.4 .8413 .8643 .8849 .9032 .9192 .8438 .8665 .8869 .9049 .9207 .8461 .8686 .8888 .9066 .9222 .8485 .8708 .8907 .9082 .9236 .8508 .8729 .8925 .9099 .9251 .8531 .8749 .8944 .9115 .9265 .8554 .8770 .8962 .9131 .9279 .8577 .8790 .8980 .9147 .9292 .8599 .8810 .8997 .9162 .9306 .8621 .8830 .9015 .9177 .9319 1.5 1.6 1.7 1.8 1.9 .9332 .9452 .9554 .9641 .9713 .9345 .9463 .9564 .9649 .9719 .9357 .9474 .9573 .9656 .9726 .9370 .9484 .9582 .9664 .9732 .9382 .9495 .9591 .9671 .9738 .9394 .9505 .9599 .9678 .9744 .9406 .9515 .9608 .9686 .9750 .9418 .9525 .9616 .9693 .9756 .9429 .9535 .9625 .9699 .9761 .9441 .9545 .9633 .9706 .9767 2.0 2.1 2.2 2.3 2.4 .9772 .9821 .9861 .9893 .9918 .9778 .9826 .9864 .9896 .9920 .9783 .9830 .9868 .9898 .9922 .9788 .9834 .9871 .9901 .9925 .9793 .9838 .9875 .9904 .9927 .9798 .9842 .9878 .9906 .9929 .9803 .9846 .9881 .9909 .9931 .9808 .9850 .9884 .9911 .9932 .9812 .9854 .9887 .9913 .9934 .9817 .9857 .9890 .9916 .9936 2.5 2.6 2.7 2.8 2.9 .9938 .9953 .9965 .9974 .9981 .9940 .9955 .9966 .9975 .9982 .9941 .9956 .9967 .9976 .9982 .9943 .9957 .9968 .9977 .9983 .9945 .9959 .9969 .9977 .9984 .9946 .9960 .9970 .9978 .9984 .9948 .9961 .9971 .9979 .9985 .9949 .9962 .9972 .9979 .9985 .9951 .9963 .9973 .9980 .9986 .9952 .9964 .9974 .9981 .9986 3.0 3.1 3.2 3.3 3.4 .9987 .9990 .9993 .9995 .9997 .9987 .9991 .9993 .9995 .9997 .9987 .9991 .9994 .9995 .9997 .9988 .9991 .9994 .9996 .9997 .9988 .9992 .9994 .9996 .9997 .9989 .9992 .9994 .9996 .9997 .9989 .9992 .9994 .9996 .9997 .9989 .9992 .9995 .9996 .9997 .9990 .9993 .9995 .9996 .9997 .9990 .9993 .9995 .9997 .9998 Also for zp = 4.0, 5.0, and 6.0, the values of p are 0.99997, 0.9999997, and 0.999999999. 14 p = P(χ2df>x2) x2 0 df χ2 Distribution 1 2 3 4 5 .25 1.32 2.77 4.11 5.39 6.63 .20 1.64 3.22 4.64 5.99 7.29 .15 2.07 3.79 5.32 6.74 8.12 .10 2.71 4.61 6.25 7.78 9.24 Tail probability .05 .025 .02 3.84 5.02 5.41 5.99 7.38 7.82 7.81 9.35 9.84 9.49 11.14 11.67 11.07 12.83 13.39 6 7 8 9 10 7.84 9.04 10.22 11.39 12.55 8.56 9.80 11.03 12.24 13.44 9.45 10.75 12.03 13.29 14.53 10.64 12.02 13.36 14.68 15.99 12.59 14.07 15.51 16.92 18.31 14.45 16.01 17.53 19.02 20.48 11 12 13 14 15 16 13.70 14.85 15.98 17.12 18.25 19.37 14.63 15.81 16.98 18.15 19.31 20.47 15.77 16.99 18.20 19.41 20.60 21.79 17.28 18.55 19.81 21.06 22.31 23.54 19.68 21.03 22.36 23.68 25.00 26.30 17 18 19 20 20.49 21.60 22.72 23.83 21.61 22.76 23.90 25.04 22.98 24.16 25.33 26.50 24.77 25.99 27.20 28.41 21 22 23 24 25 24.93 26.04 27.14 28.24 29.34 26.17 27.30 28.43 29.55 30.68 27.66 28.82 29.98 31.13 32.28 26 27 28 29 30 30.43 31.53 32.62 33.71 34.80 31.79 32.91 34.03 35.14 36.25 40 50 60 80 100 45.62 56.33 66.98 88.13 109.1 47.27 58.16 68.97 90.41 111.7 p .01 6.63 9.21 11.34 13.28 15.09 .005 7.88 10.60 12.84 14.86 16.75 .0025 9.14 11.98 14.32 16.42 18.39 .001 10.83 13.82 16.27 18.47 20.52 .0005 12.12 15.20 17.73 20.00 22.11 15.03 16.62 18.17 19.68 21.16 16.81 18.48 20.09 21.67 23.21 18.55 20.28 21.95 23.59 25.19 20.25 22.04 23.77 25.46 27.11 22.46 24.32 26.12 27.88 29.59 24.10 26.02 27.87 29.67 31.42 21.92 23.34 24.74 26.12 27.49 28.85 22.62 24.05 25.47 26.87 28.26 29.63 24.72 26.22 27.69 29.14 30.58 32.00 26.76 28.30 29.82 31.32 32.80 34.27 28.73 30.32 31.88 33.43 34.95 36.46 31.26 32.91 34.53 36.12 37.70 39.25 33.14 34.82 36.48 38.11 39.72 41.31 27.59 28.87 30.14 31.41 30.19 31.53 32.85 34.17 31.00 32.35 33.69 35.02 33.41 34.81 36.19 37.57 35.72 37.16 38.58 40.00 37.95 39.42 40.88 42.34 40.79 42.31 43.82 45.31 42.88 44.43 45.97 47.50 29.62 30.81 32.01 33.20 34.38 32.67 33.92 35.17 36.42 37.65 35.48 36.78 38.08 39.36 40.65 36.34 37.66 38.97 40.27 41.57 38.93 40.29 41.64 42.98 44.31 41.40 42.80 44.18 45.56 46.93 43.78 45.20 46.62 48.03 49.44 46.80 48.27 49.73 51.18 52.62 49.01 50.51 52.00 53.48 54.95 33.43 34.57 35.71 36.85 37.99 35.56 36.74 37.92 39.09 40.26 38.89 40.11 41.34 42.56 43.77 41.92 43.19 44.46 45.72 46.98 42.86 44.14 45.42 46.69 47.96 45.64 46.96 48.28 49.59 50.89 48.29 49.64 50.99 52.34 53.67 50.83 52.22 53.59 54.97 56.33 54.05 55.48 56.89 58.30 59.70 56.41 57.86 59.30 60.73 62.16 49.24 60.35 71.34 93.11 114.7 51.81 63.17 74.40 96.58 118.5 55.76 67.50 79.08 101.9 124.3 59.34 71.42 83.30 106.6 129.6 60.44 72.61 84.58 108.1 131.1 63.69 76.15 88.38 112.3 135.8 66.77 79.49 91.95 116.3 140.2 69.70 82.66 95.34 120.1 144.3 73.40 86.66 99.61 124.8 149.4 76.09 89.56 102.7 128.3 153.2 15 
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