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Unformatted text preview: IME602: Probability and Statistics
Quiz4, October 27, 2016 An A c0 t‘< Duration: 1 hour Maximum Marks: 10 Name: ....................................... Roll No: This is a closed book/note examination. Useful information, whenever necessary, is provided
along with the question. Attempt all questions. 1. Let X1 and X 2 denote your quiz1 and 2 marks. Let us assume X1 and X 2 to be independent
and uniformly distributed in [0,10]. Let Y = X1 + X 2 denote the total marks in quiz1 and 2.
Derive the density function of Y. Represent fy (y) pictorially. [Marks: 3] Info: If X1 and X2 are independent, fx1+x2 (y) = L: fx1 (x) sz (y — x)dx. 2. Let C denote the melting point of a newly developed material. Measurements of c, due to
error, are of the form: Mi = c + 61, where ei’s are the errors. Let us take ei’s to be iid random
variables with mean 0 and variance 3. Then Mi’s are iid random variables too. We consider IV! = Z?=1Mi/n as the measurement for c. Using Chebyshev’s inequality, in quiz3, we got
11 2 120 to be 90% sure that [V] is in c i 0.5. Using the central limit theorem, ﬁnd the new
minimum of n to ensure the same accuracy. [Marks: 2] Note: According to the central limit theorem, X = Z?=1Xi/n, where Xi’s are iid random
variables with mean ,u and variance 02, approximately follows N (u, 02 / n) for large n. 3. We want to estimate the distribution function value of an arbitrary random variable at an
arbitrary point, i.e., we want to estimate 9 = Fx(a) for any X and a. We have a random
sample of X, denoted by (X1,X2, , Xn). Then it makes sense to estimate 0 by the proportion of Xi’s that do not exceed a, i.e., 0 = (221:1 1x,5a)/na where 1Xi5a = 1 if X, S a, otherwise 1X,sa = 0. Study the properties of g (as an estimator of 0). [Marks: 2.5] Hint: 1xi5a’5 are iid random variables. Identify their mean and variance ﬁrst. 4. Estimate k and p for N e gB in(k, p) using the method of moments. [Marksz 2.5] Hint: Geometric(p) has mean 1 / p and variance (1 — p) / p2. I am talking about geometric
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 Fall '17
 Faiz Hamid
 Probability theory, 5%, 5°, Mama, Ilmdrtw

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