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midterm1sol - Midterm Examination 1 Sta 113 Probability and...

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Unformatted text preview: Midterm Examination # 1 Sta. 113: Probability and Statistics in Engineering ‘ Tuesday, 2009 Oct. 13, 1:15 — 2:30 pm This is a closed—book exam so do not refer to your notes, the text, or any other books (please put them on the floor). You ma}r use a single sheet of notes or formulas and a calculator, but materials may not be shared. A formula sheet and four blank worksheets are attached to the exam. You must Show your work to get partial credit. Even correct answers will not receive full credit without justification. Please give all numerical answers to at least two correct digits or as exact fractions reduced to lowest terms. Write your solutions as clearly as possible and make sure it’s easy to find your answers (circle then) if necessary), since you will not receive credit for work that I cannot understand or find. Good Luck! If you find a. question confusing please ask Ine to clarify it. Cheating on exams is a breach of trust with classmates and faculty, and will not be tolerated. After completing the exam please acknowledge the Duke Honor Code with your signature below: I have neither given nor received unauthorized aid on this exam. Signature: Print Name: ARSMGZM toy Name: M Sta 113 Problem 1 : For each problem select an answer and state why. (15pts) i. ii. iii iv . V. F all The maximum likelihood estimator and the maximum a posteriori estimator are equivalent when the prior is uniform. @. Event A occurs with probability 0.4. The conditional probability that A occurs given that B occurs is 0.2, while the conditional probabil- it},r that A occurs given that B does not occur is 0.7. Vi’hat is the conditional probability that B occurs given that A occurs? ./10 (193/? (c)5/6 (aw/10 . An unbiasecl estimator is always the best estimator. (a) T @F The least squares estimator is the MLE (maximum likelihood estima- tor) for a linear function with what type of noise distriution (a) x2 (b) log—normal (c) Poisson‘ (d none of the above If X1, ...,X,1 is distributed as iid Exponential distributions with pa— f“ eter A then the mle estimate of A is biased. ( a) (b) F 2009 Page 1 of 10 Oct. 13, 2009 Q) Azaxm‘jl #53373 With/em : S (xfilc‘xeb'; 8 S (“3345* -. pr Name: w Sta 113 Problem 2: . A bank operates both a drive-up facility and a walk-up Window. On a randomly selected day, let X = the proportion of time that the drive-up facility is in use and Y z the proportion of time that the walk~up window is in use. Then the set of possible values for (X , Y) is the rectangle D = {(33, y) : 0 g :1: 5 1,0 5 y S 1}. Suppose the joint pdf of (X, Y) is given by , n 55+?! 055831351151 f(a,,y)~{ 0 otherwise (15 points) a. Are X and Y independent? b. If X is found to be 0.5, what is the probability that the walk—up window is busy at most half the time? c. What is the probability that the use of the two facilities combined is less than or equal to .7 proportion of time. Fall 2009 Page 2 of 10 Oct. 13, 2009 .‘7 ,7—x O O .ll‘t3 H Name: —_._____ Sta 113 Problenl 3 z “’e are given the following exponentiai model 2 :T x 6‘”, where log(7') ~ \To(p, 02). (25 points) :1. Given observations (21,311), ..., (2,1,3JH) drawn iid from the exponential model use MLE to estimate c1. b. Given prior “(01) = wJ-lé—gefici/g provide the MAP estimate. (3. Provide the Bayes estimate (you can leave this in integral form). .: 0,06: + e,><'L / gram: A, N (we?) Fall 2809 Page 3 of 10 Oct. 13, 2009 S at we.) L024) do, “WWW... ,_.,___=,‘ STU“) l—lCI) do! Name: Ste 113 Probiem 4: The NY times book list contains 120 books. Of these 80 are fiction and 40 are nonfiction. You buy 20 different books at random from the times book list. (20 points) a. Compute the probability density function of the number of fiction books. b. Suppose you didn’t know how many fiction books were in he list. You observe that you bought I; : 13 fiction books of the 20. Can you compute PIUM | k 2 13), where PHI is the number of fiction books. (Hints: Use Bayes’ Rule. Yo ou don t need to compute the number, the fonnula 01 equation is enough.) a) W: o : 47>) b) thltm “'2 B) 1 12.6 Fall 2009 Page 4 of 10 Oct. 13, 2009 Name: M Sta 113 Problem 5 : X1, ...,Xm are drawn iid from a binomial distribution with parameters 1:» and n. For the remainder of the problem it is a fixed and known parameter. (25 points) a. State the likelihood for X1, ..., Xm. 1). Compute the maximum likelihood estimate of p. 6. Given the above estimate of 13 use the central limit theorem to provide a density for 13 at) Lei: M (3;) Wit?) V\-—X[ A‘ W :27) F“ WM A : j: '2: @9413: LBJ“) ? "m MP?) A - ml 1;[:(th):« Mn \Vhfl» mama M Fall 2009 Page 5 of 10 Oct. 13, 2009 ...
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