553311Summer2024Midterm2 (1)
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Johns Hopkins University**We aren't endorsed by this school
Course
EN.553 300
Subject
Statistics
Date
Dec 17, 2024
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9
Uploaded by HighnessHerring4859
Exam IIName:Section:Johns Hopkins UniversityEN.553.311Intermediate Probability and StatisticsSummer 2024
Do Not Write on this PageInstructions:Try your best to complete the following problems. Justify each solution, or it will not receive credit(even if it is correct). You may tear off the last page for scratch work. Cheating will result in a grade of 0 and anacademic integrity violation will be reported.Throughout,N(µ, σ) denotes the normal distribution with meanµand standard deviationσ.ProblemScoreP1/20P2/20P3/20P4/20P5/20P6/04Total:/1001
P1.If Var(Y−X)>Var(X) + Var(Y) thenXandYare. . .(a) positively correlated;(b) negatively correlated;(c) not correlated.Choose one and justify your answer:P2.LetX1∼ N(1,2) andX2∼ N(3,4) be independent random variables. LetW=eX1andY=eX2. ThenE(W2Y3)=?2
P3.It is a well known fact that cats have 9 lives. Suppose that each lifetime of a cat follows a normal distribution withmean 16 years and standard deviation 2 years (independently of any other lifetime). Compute the probabilitythat a randomly selected cat will live for at least 153 years over all 9 of their lives.P4.A father buys packs of batteries for his child’s remote control car. Assume a single pack of batteries will last forXdays, whereX∼Gamma(16,3). Approximately how many packs of batteries should the father buy so thatP(¯X <49)= 0.8413?Reminder:The expectation and variance of a Gamma(α, β) random variable isαβandαβ2, respectively.3
P5.Suppose Rick Barry attempted an unknown number (sayn) free throws at basketball practice. It is known thatRick makes each free throw attempt with probabilitypindependently of any other attempt.(a) If Rick madexfree throws at practice, compute ˆnmme, the method of moments estimator of the numberof free throws Rick attempted.(b) Compute ˆnmle, the maximum likelihood estimator for the number of free throws Rick attempted.[Hint:Consider the ratioL(n)/L(n−1).]4
P6.What is your favorite movie?5
Scratch WorkThis Page Will Neither be Read nor Graded.6
Scratch WorkThis Page Will Neither be Read nor Graded.7
648Appendix of TablesTABLE A1Standard Normal Distribution Function:(x)=1√2πx−∞e−y2/2dyx.00.01.02.03.04.05.06.07.08.09.0.5000.5040.5080.5120.5160.5199.5239.5279.5319.5359.1.5398.5438.5478.5517.5557.5596.5636.5675.5714.5753.2.5793.5832.5871.5910.5948.5987.6026.6064.6103.6141.3.6179.6217.6255.6293.6331.6368.6406.6443.6480.6517.4.6554.6591.6628.6664.6700.6736.6772.6808.6844.6879.5.6915.6950.6985.7019.7054.7088.7123.7157.7190.7224.6.7257.7291.7324.7357.7389.7422.7454.7486.7517.7549.7.7580.7611.7642.7673.7704.7734.7764.7794.7823.7852.8.7881.7910.7939.7967.7995.8023.8051.8078.8106.8133.9.8159.8186.8212.8238.8264.8289.8315.8340.8365.83891.0.8413.8438.8461.8485.8508.8531.8554.8577.8599.86211.1.8643.8665.8686.8708.8729.8749.8770.8790.8810.88301.2.8849.8869.8888.8907.8925.8944.8962.8980.8997.90151.3.9032.9049.9066.9082.9099.9115.9131.9147.9162.91771.4.9192.9207.9222.9236.9251.9265.9279.9292.9306.93191.5.9332.9345.9357.9370.9382.9394.9406.9418.9429.94411.6.9452.9463.9474.9484.9495.9505.9515.9525.9535.95451.7.9554.9564.9573.9582.9591.9599.9608.9616.9625.96331.8.9641.9649.9656.9664.9671.9678.9686.9693.9699.97061.9.9713.9719.9726.9732.9738.9744.9750.9756.9761.97672.0.9772.9778.9783.9788.9793.9798.9803.9808.9812.98172.1.9821.9826.9830.9834.9838.9842.9846.9850.9854.98572.2.9861.9864.9868.9871.9875.9878.9881.9884.9887.98902.3.9893.9896.9898.9901.9904.9906.9909.9911.9913.99162.4.9918.9920.9922.9925.9927.9929.9931.9932.9934.99362.5.9938.9940.9941.9943.9945.9946.9948.9949.9951.99522.6.9953.9955.9956.9957.9959.9960.9961.9962.9963.99642.7.9965.9966.9967.9968.9969.9970.9971.9972.9973.99742.8.9974.9975.9976.9977.9977.9978.9979.9979.9980.99812.9.9981.9982.9982.9983.9984.9984.9985.9985.9986.99863.0.9987.9987.9987.9988.9988.9989.9989.9989.9990.99903.1.9990.9991.9991.9991.9992.9992.9992.9992.9993.99933.2.9993.9993.9994.9994.9994.9994.9994.9995.9995.99953.3.9995.9995.9995.9996.9996.9996.9996.9996.9996.99973.4.9997.9997.9997.9997.9997.9997.9997.9997.9997.9998