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Unformatted text preview: [24} Indian Institute of Technology, Kharagpur Date of Exam.: .1109 (EN/AN) Time: 3 Hrs. Full Marks: 50 No. of Students: 50 End (Autumn) Semester Examination (2009—10) Department: Mathematics Subject N0. MA41009 Subject Name : Probability & Statistics 1st yr of two year M.Sc. (Mathematics) / 2nd yr of integrated M.Sc. (Economics) Note: Use of calculator and Statistical tables is allowed. Question 1 is compulsory. Answer any 7 questions from 2 to 9. 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 answerscripts. (a) Suppose X~ N(u, 02). The pdf of Y = Z2 is given by ................. .. (b) Let 32 be the sample variance of a random sample of size 26 fromlN(,u.,16). The value of P(9.35 < 5’2 < 24.09) is .............. .. (e) Let X5, z' = 1, 2, . . . , 10 be independent random variables. Each X,- is an uniformly distributed random variable over [0, l]: The approximate value of P(E%£1X§ > 6) (Using central limit theorem) is ......... .. ((1) Let X5, 1' = 1, 2, . . . ,n be independent random variables and Xg follows exponential distribution with mean 1. The moment generating function of W = 23:11:} is (e) Choose the correct answer: The critical region is the region where the null hypothesis is ........ .. (rej ected / accepted) . (f) Suppose X~ N(,u, 02) and both it and 02 unknown. Classify each of the hypotheses simple or composite Ho : ,u. = 10, o'2 = 166i) Ho : n > 10, o2 = 16(2'122') H0 : a > 10. (g) Let a random sample of size 16 from N(,u, 02) yield E = 4.7 and 52 2 5.76. The 90% confidence intervallfor a is ....... .. (h) Let X m 3(500, 0.80}. The approximate value of P(375 s X S 425) is....... (i) Suppose that that ductile strength X of a material has a lognormal distribution with parameters ,u = 5 and or = 0.1. The value of P(110 S X S 130) is Suppose the distribution of scores on an IQ test has mean 100 and ad 16. The least value of probability of a student having an IQ above 148 or below 52 is at most........ [10x1.5] 2. (a) The time to failure in months, X, of the light bulbs produced at two manufacturing plants obey exponential distributions with means 5 and 2 months, respectively. Plant B produce three times as many bulbs as plant A. The bulbs are indistinguishable to eye intermingled and sold. What is the probability that a bulb purchased at random will burn at least 5 months. (b) Let X1 ~ N(3, 9), X2 ~ N(0, 1) and X3 ~ N(1, 1) are independent random variables. Further let Z] = 2X1 + 3X2 —~ 4X3 and Z2 = 3X1 — 2X2 + X3. Find the correlation coefficient between Z1 and Z2. (c) It is known that the probability of a user’s being able to log on a computer from a remote terminal at any given time is 0.7. Suppose that if user has not been able to log on in 4 attempts, then what is the probability that lie-will need 3 more attempts to be able to log on. (d) Let the random variable X~P(.\) (Poisson random variable with parameter A). Find the probability that X takes even numbers {0, 2, 4, ...., }. [5] 3. Let the random variables X and Y have joint pdf fx,y(a:,y) = Say, 0 s y S a: 5 l. (a) Is the function given above a pdf? Determine the marginal (fx(:c), fy(y)) and con- ditional (fX|Y($lylafv|x(yl-Tll Pdf’S- (b) Are X and Y independent? (0) Compute the P(Y < 1/8|X = 1/2). [5] 4. (a) Given the random variables X and Y with their joint probability distribution as X \ Y -1 1 an (1) Find Cov(X,Y) (ii) Are X and Y independent? (b) Let the (X, Y) ~ BVNmm, My, 03,, 0:, ,0). Determine the conditional pdf of Y given lev|x(yl$)l- (e) Let us assume that the distribution of grades for a particular group of students, where X and Y represent the grade point average in high school and the first year college, respectively, follow a bivariate normal distribution with parameters an, a 3.2m]; = 2.4, om x 0.4, 03. = 0.6 and ,0 = 0.6. Evaluate the following probabilities: (i) P(Y < 1.3) (a)P(Y < 1.8|X = 2.5). [5] 5. (a) Find the mean and variance of the lognormal distribution with parameters ,u and :72. (b) Given 10 iid 1'.vs. X5, where X,- ~ N(0, 25) for t' = 1,2,... , 10. Find the number k such that 1 10 _. 2> =, P(102X,_k) 05 1‘21 (c) The tensile strength (X) for a type of wire is normally distributed with an'unknown mean ,u and unknown variance 02. Five pieces are randomly selected from a large roll, and strength of each segment of the wire is measured. Find the probability that the sample mean T will be within of the true population mean a. ( 5'2 is sample variance). (a) Let X1, X2,X3, X4 be a random sample from N(O, 02). Find the distribution of r.v. W = % with suitable parameter(s). [5] 6. The time a client waits to be served by the mortgage specialist at a bank has density function 1 = gage—3’16 31' > 0,6 > 0 (a) Derive the maximum likelihood estimator of :9 for a random sample of size it. Is the (h) (c) 7. (a) 00} (c) 8. (a) (b) (c) 9. (a) (b) estimator unbiased. Find the variance of this estimator. Derive the method of moment estimator of 6. If the waiting times of 15 clients are 6, 12, 15, 14, 12, 1'0, 8, 9, 10, 9, 8, 7, 10, 7 and 3 minutes, compute the maximum likelihood estimate of 6. [5] A random sample of 30 items is taken from a normal distribution with unknown mean p. and ad. or z 2.5. Given that 2?; I; m 77, calculate the 90% confidence interval for the population mean. Suppose that a. random sample of size n is taken from a normal distribution with unknown mean a and ad. cr = 5. Calculate the the minimum sample size so that one can be 95% confident the interval [if — 1,? + 1] contains the true value of n. A random sample of 15 items is taken from a N (smog) population that yielded i=1 xi = 53 and at? = 222. Another random sample of size 11 is taken from a N(ny, 0'3) population independent of the first sample such that 2%; y,- = 77 and 2}; yf = 560. Obtain a 95% confidence interval for (an: — pay) by assuming the true but unknown variances are equal. [5] An experiment consists of 12 independent Bernoulli trials. If H0 : p = 1/3 is tested against H1 : p > 1/3 by using the decision rule, “ reject Ho, if y, the number of successes equals or exceed 8.” Find 0:. Also find )3 if p a 1 / 2. Independent random samples of sizes 711 100 and in = 100 were selected from N(,u1, 012) and N(,u.2, 0%), respectively. The samples yielded following informatibn: if = 18.7.3? = 351,172 = 16.4,33 = 58.2 If your objective is to show that m is larger than #2, state the null and alternative hypotheses. (ii) Is the test in part one- or a two- tailed test? (iii) Give the test statistic that you would use for the test in part (i). (iv) State the underline assumptions (v) Calculate the value of the test statistic. (vi) Draw your conclusion at or = 0.05. ' [5] Let U1, U2, . . . , Un are n independent r.vs. with U;- distributed as gamma (1/2, 1/2). Find the mgf of W = 23:1 U5. Identify the distribution of W with suitable parameter _ and write down its pdf and denote it by fw(w). Let Z ~ N(O, 1) and W N fw(w). Assuming Z and W are independent. find the pdf of the random variable T = [5] Discuss the desirable properties of the estimator. ...
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