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Unformatted text preview: Indian Institute of Technology, Kharagpur Date...................... FN/AN Time: 3 Hrs Full Marks: 50 No. of Students: 40 END (Autumn) Semester 2010-11, Deptt: MA Sub. No. MA41009 Subject Name: Probability and Statistics ' Note: Use of calculator and Statistical tables is allowed. Answer all questions. 1. Examine each part of this question and carry out your solution as usual. Write ONLY THE ANSWER for each part of this question on the first page of your answer- script. (a) Suppose the probability that an engine of an automobile does not work during any one hour period is 0.02. Find the probability that a given engine will work for 2 hours. (b) Of the peeple passing through an airport metal detector, 0.5% activate it. Let X: the number among the randomly selected group of 500 who activate the detector. Compute the P(X 2 5). (c) A random variable X. has mean 24 and variance 9. Obtain a lower bound on the probability that the random variable X assumes value between 16.5 to 31.5. (d). If X ~ N(1,4) then find P(3 < X < 5). (e) Suppose X1 and X2 are independent identically distributed standard normal (i.e. N0), 1)), random variable Then what is the distribution of X1— X2? (i) Let X, Y and Z are three uncorrelated variable having variance 0%, a; and a: respectively. What is the correlation between U— X + Y and V: Y + Z '? (g) Let X1,X2,X3,X4 is a random sample of size 4 from N(100, 25). What is the distribution of Y = X1 — 2X2 + X3 — 3X4? (h) A random sample of size 18 from a normal population with both mean p. and I variance 0'2 unknown yield 5' = 2.27 and sample variance 32 = 1.02. Determine a 99% confidence interval for the actual variance 02. (i) Let X ~ N (a, 03). Classify the following hypotheses as simple/composite. (1915'an10,02>4(it)H:,u=10,02#4(iii)Ho:n=10,02=4(iv) H. ‘ a = 10 (j) Which one of the following is true in case of (A) Type I error, and (B) type _II ' error. (i) Reject Ha I H1 is true (it) Reject HD | H0 is true (iii) Accept Ho | H0 13 true . (to) Accept Ho | El is true, ' _ [1x10] 2. (a) Suppose that a bomber is making attempts repeatedly to hit a target which is a square of 40 ft by 40 ft. with center at the origin and'sides parallel to the coordinate axes. Denote by (X, Y) the actual point hit. Assume X and Y are . independent Normal with mean at the origin and variance 400 ft. sqfift. (1") Find the prob ability that the target is hit for the first time at the 10“” attempt. (it) Find the probability of getting at least one hit in 10 attempts. Given so) = .8418. - -'—--P.T.O (b) Suppose a point is chosen at random from (-3, 2). Let X be the random variable denoting the absolute value of the point chosen, i.e., X (11)) = [w|, w E (-3, 2) Find the distribution function of X. [5+3] 3. (a) Suppose the bivariate probability distribution of a random vector (X, Y) is given according to the entries in the following table. nun-El “In“?! "mun “Ian“ (1) Compute the conditional probability function of X given Y = 2. (it) Compute marginal probability function of X. (iii) Are X and Y independent? (b) Suppose that a bivariate random vector (X, Y) has the joint probability density function ' 2 6‘9”?) if 0 < any < oo f(a:,y) = { 0 otherwise (2') Compute the marginal p.d.f fx(:r) of X. (ii) Calculate P(X < Y). (G) Let X be a continuous random variable with distribution function 13(3) given by HIP- 0 ' .7: < 0 §[1+in( )] 0<x<4 1 I :z: > 4 (1) Obtain the pdf fix) (ii) Compute P(1 g X g 3) [2+3+3] 4'. (a) Suppose the time X between orders for a given part, at a large warehouse is a gamma (r, A) r. v with pdf . _ X. r-l “Am f(:s .,r,A).~ “was s $>0. (3') Obtain the moment generating function of Y = 2AX. (ii) Find the method of moment estimators of r and A. If the 10 observations yield valties: 15.5, 4.5, 6.8, 46.0, 34.5, 4.7, 20.9, 8.2, 14.9, 17.7, then'obtain estimates - of i" and A. (b) Let X1,-..X2, WX be a random sample of size n from a population with the probabilty density function f(x, 6): 56 "Ix 9' ,-oo<a:<oo,—oo<6<oo I Estimate. the parameter [5' by the method of maximum likelihood. [5+3] ' -—-P. T. O - 5. The following information was obtained from two independent samples selected from two normally distributed populations: WEI-- (a) Calculate sample means 3*." and "y‘, and sample variances 3E and 6%. (b) Test Ho : of = 0% against H1 : a? 75 0%. (a = 0.05) (c) Based on the decision in (b), use a suitable test for testing Ho : #1 = it; against a :m as #9- (a = 0-05) (d) Discuss the desirable properties of an estimator. [2+2+2+2] 6. (a) Let X1,X2,X3 be a random sample from a distribution with unknown mean a and known variance 02. Let T = Y and W = (2X1 +X2 +5X3)/8 are estimators of the population mean {4. Are T and W unbiased estimators? Which one of the two is best estimator and why? (b) Let X1, X2, . . . ,Xfl be a random sample of size n=25 from normal distribution with an unknown mean p. and variance (72:16. We wish to test the hypothesis H, : u = 10 against H1 : p‘. > 10. Let the rejection region is defined by : Reject Ha if the sample mean 3: > 11.2. Find 0:. Also find power of the test if H1 :n = 11. ((3) Suppose we want to know the effect on driving of a drug for cold ”allergy, in a study in which same peeple were tested twice, once after 1 hour of taking the ' ' drug and once when no drug taken. Suppose we obtain the following data, which represent the number of cones (placed in a certain pattern) knocked down by each of the nine individuals before taking the drug and after an hour of taking the drug . mum-Inn.- manna Assuming that the difference of 'each pair is comming-from an approximately normal distriblition, test if there is any difference in the individuals’ driving ability - under the two conditions. Use a = 0.05. [2+3+3] . The End ...
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