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midterm_1sol

# midterm_1sol - Midterm Examination#1 Sta 113 Probability...

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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 may 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 Inc 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: Anst—u- law Name: _________—_— Sta 113 Problem 1 : For each problem select an answer and state why. (15pts) i. The maximum likelihood estimator and the maximum a posteriori estimator are equivalent when the prior is uniform. o M ii. 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. What is the conditionai probability that B occurs given that A occurs? ./10 (b) 3/7 (c) 5/6 ((1)7/10 iii. An unbiased estimator is always the best estimator. (a) T @F iv. The least squares estimator is the MLE (maximum likelihood estima— tor) for a linear function with what type of noise distrintion (a) X2 (b) log—normal (c) Poisson @ none of the above v. If X1,...,Xn is distributed as iici Exponential distributions with pa- r eter A then the nile estimate of A is biased. ti" F Fall 2009 Page 1 of 10 Oct. 13, 2009 Name: w Sta 113 Problem 2: A bank operates both a drive-up facility and a. walk—up window. 011 a randomly selected day, let X = the proportion of time that the (hive—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 = {(\$,i ) '. 0 S :1: s 1,0 5 y S 1}. Suppose the joint pdf of (X,i’) is given by ,. _ \$4.31 03351332151 f(1b:y)*{ 0 otherwise (15 points) a. Are X and Y independent? 1). If X is found to be 0.5, what is the probability that the walk—up window is busy at most half the time? o. What is the probability that the use of the two facilities combined is less than or equal to .7 proportion of time. 0‘) {(X): ”twain 1* (X3 +%Z) Ln»: heavsXZa+t ‘7’ ‘ 1 ~ x+_ O 2 0 Shot QUAD ~75 {’00 We) , NOT WDEW—‘me'ﬂlﬁ e ithfi: ﬁgﬁg =s Po<5vmo=j Fall 2009 Page 2 of 10 Oct. 13, 2009 c) ArEXIJ: “33"“ 7 7- MM’eA) :.. S (xmc‘mb‘; i S (xbyajax a A 0 O . H43 Name: @h—mw_ Sta. 113 Problem 3 : We are given the following exponential model :5 = T x 66193, where logh) ~ Nounaz). (25 points) a. Given observations (21,3;1), ..., (211,332) drawn iid from the exponential model use MLE to estimate cl. b. Given prior 1r(c1) : ﬁfe?” provide the MAP estimate. (3. Provide the Bayes estimate (you can leave this in integral form). i: t + 0“IKL / £61? A, N (“’61) . 2. "2"“ I @n2i1(%?‘”ik”a‘x‘ E ﬁﬁ'w A 7. i Z ‘3‘” WQHQ {A (7 ._ .____ \ = we” We ‘ ‘* Fall 2009 Page 3 of 10 Oct. 13, 2009 Sc. Tim Lox!) do, "MM—mm THC‘) L (CE) Cl C,’ Name: w Sta 113 Problem 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 1.: = 13 ﬁction books of the 20. Can you compute Pl‘(ﬂ€[ { I»: Z 13), where 1W is the number of ﬁction books. (Hints: Use Bayes’ Ruie You don t need to compute the number, the f01 mula or equation is enough.) a) Merle) a @Qij‘m M mom‘M , b) Nuwu—ﬁ) rim)“ 7 MM Whig Fall 2009 Page 4 of 10 Oct. 13, 2009 2‘ (TEM‘ZTWW Name: M Sta 113 Problem 5 : X1, ...,Xm are drawn iid from a binomial distribution with parameters p and n. For the remainder of the problem n is a ﬁxed and known parameter. (25 points) a. State the iikelihood for X1, ..., X m. b. Compute the maximum likelihood estimate of p. o. Given the above estimate of 13 use the central limit theorem to provide a density for 13. me \p A f H:(:Xt I ?(nP) \Wﬂ‘ “T: M W‘ Fall 2009 Page 5 of 10 Oct. 13, 2009 “—1); ...
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midterm_1sol - Midterm Examination#1 Sta 113 Probability...

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