CS MAINS STATISTICS Paper-I
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Time Allowed: Three Hours
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Maximum Marks: 250
SECTION-A
Q. If for the events A and 0 and 0 then show that p(A if and only if Likewise, if and only if
Q. Three digits 1,2 and 3 are written down in random order; What is the probability that at least one digit will occupy its proper place? 10
Q. Construct a 95% confidence interval for I.l in N(l.lt .4) from the following observed sample: 15. 10
Q. Let Xl and X2 be i.i.d. random variables and each has probability density function <img src='./qimages/298-1-d.jpg'> otherwise. If U1=0.6 X1+0.4 Xl and V2 =XI X2, decide which one of U1 and U2 are sufficient statistics for 9. If then show that it has smaller variance compared to Var(U1). 10
Q. A tourist resort is visited by tourists from many countries. The resort operator has the following data in the month of January, 2014 Country USA UK Canada Italy Germany France Japan No. of tourists 22 12 18 10 20 18 30 The resort operator has a hypothesis that the proportion of tourists visiting in the month of January of any year is 2 1 2 I 2 2 3. Test" this hypothesis at level of significance (Given 0.05 12.59). 10
Q. 2(a)The probability density function of a random variable X is given by fx(x)=630x4(1 x)4 for 0 x otherwise. Find the probability that X will take on a value within two standard deviations of the mean (u± 2cr) and compare it with the lower bound provided by the Chebychev's Inequality.20
Q. For a random variable X with probability density function <img src='./qimages/298-2-d-1.jpg'> otherwise; where y>0 is. an integer, show that <img src='./qimages/298-2-d-2.jpg'>
Q. A die is rolled 15 times with the following results Face value: 1 2 3 4 5 6 Frequency 1 4 0 4 6 Use Kolmogorov-Smirnov statistic to test whether the die or not. (Given DI5;0.05=0.304) 15
Q. Suppose n items are put on test simultaneously and the test is continued until r items fail. . Assuming an exponential failure distribution with mean life time obtain the maximum likelihood estimator for and hence estimate Also obtain the Fisher information . about the parameter e and show that the estimator is asymptotically normal. 20
Q. Let and the prior distribution of u is N 1). Assuming squared error loss function, obtain the Bayes estimator for J.l. Also obtain Bayes risk. 15
Q. Define convergence in probability and convergence in distribution of a sequence of random variables. Show that convergence of Xn to X in probability implies convergence of Xn to X in distribution. Is the converse also true? 15
Q. Let Xn be a random sample drawn from a uniform distribution. Obtain a.UMP test of size a for testing Ho:0=00 against HI 20
Prove the following results A sequential probability ratio test (SPRT) always terminates with probability one. For a SPRT with Z=log <img src='./qimages/298-4-b-2.jpg'> such that then prove that <img src='./qimages/298-4-b-2.jpg'>
Q. Let the probability function of X and y be <img src='./qimages/298-4-c.jpg'> Prove that 15
SECTION-B
Q. 5(a)Let y3 be three independent observations having expectations <img src='./qimages/298-5-a.jpg'> Obtain least square estimates of Po, PI and Can you obtain unbiased estimate of cr'l Justify your claim. . 10
Q.5(b) If pie1 and pie2 denote two bivariate normal populations where and <img src='./qimages/298-5-b.jpg'> .Compute the Mahalonobis. distance between pie1 and 1pie2. 10
Q.5(c) For a population size 10, show that in SRS without replacement the probability of drawing a specified unit at 5th draw is equal to the probability of drawing it at the first draw.10
Q. Consider a RBD with 4 treatments arranged in 5 blocks. Show that RBD is orthogonal.10
Q.5(e)Let X2, X3, X4 and X5 be independent and identically distributed random vectors with mean vector u and covariance matrix sigma. Find the distribution ofY =XI X2+ X3- X4
Q. let whwre <img src='./qimages/298-6-a.jpg'> Obtain the conditional distribution of given 20
Q. Consider the area under wheat for a sample of 44 clusters selected from i 1 different villages. Four clusters were selected from each of the 11 villages and each cluster consists of 8 consecutive survey numbers (fields). Here sum of squares due to between villages is 2000, SS due to between clusters within villages is 8250 and Total SS is 30000. Write the ANOVA table. 15
Q.6(c) Construct an LSD of size 5. Delete one column from this LSD. Prove that the resulting design is a symmetrical BIBD with parameters v b r k 4 and A. 3. 15
Q. Explain Horvitz-Thomson, estimator the population mean. Show that H-T estimator is unbiased estimator of population mean. Also obtain its variance. 20
Q. Define Hotelling T2. State, anyone of its applications. A random sample of size 3 from bivariate normal sigma) distribution gave following unbiased estimates of and sigma <img src='./qimages/298-7-bjpg'> Compute T2-statistic to test the hypothesis H0 15
Q. Construct a key block of 27 confounded factorial experiment into a block of size 8 by choosing suitable interactions only. Write all the independent and generalized confounded interactions. Further obtain two or more blocks using key block. 15
Q. Consider a BIBO with parameters v 11 r k 5 and A 2. Obtain c matrix of this design and its non-zero eigen value. Hence obtain <img src='./qimages/298-8-a.jpg'> 20
Q. 200 boys and 100 girls of a college appeared in an examination. Means and variances of their scores are as given below Category No. of Students Mean Marks Variance Boys 200 40 10 Girls 100 50 20 How will you draw a random sample of size 30 using proportional allocation Hence obtain the variance of the estimator of the population mean. 15
Q. In the Gauss-Markov set up Xp, a least square estimate is given by a solution
of the system
X'xbeta=X'Y
Justify the above statement.
Establish that the above system of equations is always consistent.