QP-IFSM-25-STATISTICS-PAPER-I-241125

Indian Forest Services (Main) Examination-2025 DJSM-U-STSC STATISTICS PAPER—I Time Allowed : Three Hours Maximum Marks : 200 QUESTION PAPER SPECIFIC INSTRUCTIONS Please read each of the following instructions carefully before attempting questions There are EIGHT questions in all, out of which FIVE are to be attempted. Question Nos. 1 and 5 are compulsory. Out of the remaining SIX questions, THREE are to be attempted selecting at least ONE question from each of the two Sections A and B. Attempts of questions shall be counted in sequential order. Unless struck off, attempt of a question shall be counted even if attempted partly. Any page or portion of the page left blank in the Question-cum-Answer Booklet must be clearly struck off. All questions carry equal marks. The number of marks carried by a question/part is indicated against it. Unless otherwise mentioned, symbols and notations have their usual standard meanings. Assume suitable data, if necessary, and indicate the same clearly. Answers must be written in ENGLISH only. DJSM-U-STSC/58 1 [ P.T.O. (nisM) esoived 129103 nsibn! ChCnoltenimeya ‘seorow As 1. (a) There are sometimes errors in transmission of coded messages. Morse code uses ‘dots’ and ‘dashes’ which generally happen in the proportion 3: 4. Assume that there is interference on transmission line and a received dot is mistakenly taken as a dash with probability 2 and vice versa. Find (i) the probability that a dot has been received and (ii) the probability that a dot was sent given that dot was received. 4+4=8 (b) —() Check whether fx <0 2 1 F(x) = > , O<x<l 1x, 1-2, x21 2 is the cumulative distribution function of a random variable X. (i) For i=1, 2, ..., n, let X;’s be independent and identically distributed as Nu, 67). Show that weak law of large numbers holds. 4+4=8 (c) Let X,, Xo,..., X, be a random sample from U(0, 8), 8€@=(0, ~). Let n Xin) = max {X,, Xo, ..., X,} and Y, =2X;. i=1 (i) Show that Xin) is consistent estimator of 6. (ii) Check the estimator Y, for consistency and unbiasedness for the parameter 98. 4+4=8 (d) Let X,, Xo, ..., X, be a random sample from exponential distribution with mean 8. () Obtain the maximum likelihood estimator of 8. (i) Obtain the asymptotic distribution of the maximum likelihood estimator that you have obtained in (i) above. 4+4=8 (e) Let X be a random variable whose p.m.f. under Hy and Hy, is given by x 1 2 3 4 5 6 7 f(x|Ho) 0-01 0-01 0-01 0-01 0-01 0-01 0-94 f(x|Ay) 0-06 0-05 0-04 0-03 0-02 0-01 0-79 (J Find the most powerful test for Hy versus H, with size 0-04. Compute the probability of Type II error for this test. (i) Write down the most powerful test for Hy versus H, with size 0-10. Compute the power of your test. 4+4=8 DJSM-U-STSC/58 2 2. (a) Let Ee -l<x<l f(xy=} 2 O , otherwise be the probability density function (p.d.f.) of a continuous random variable X. Let Y = X?. Find (i) the probability density function of Y and check that it is a valid p.d.f., (ii) E(Y) and (iii) V(Y). 4+3+3=10 (b) () Let {X,} be a sequence of random variables such that 1 P(X, =0)= 1-(3)" and Then check whether X,, — X in probability, almost everywhere and in second mean, where X is a random variable taking value ‘0’ with probability 1. (ii) State Glivenko-Cantelli theorem. (iii) Let {A,} be an independent sequence of events such that YPAn) =< n=l Then show that r[timsupa, = P(A, io) =1. 6+44+5=15 n (c) () Let X,, Xo, .., X, be a random sample from normal distribution with mean @ and variance 6”, where o” is known. Obtain the uniformly most powerful size o test for testing Hy : 8 <8 versus H, : 8 > 89. Using the test, write down (1-«@) level one-sided confidence interval for 9. (ii) State the purpose of Wilcoxon-Mann-Whitney test and write down the test statistic. State the asymptotic distribution of the test statistic. 8+7=15 3. (a) For two continuous random variables X and Y, the joint probability density function of (X, Y) is given as e(x+2y), O<x<2, O<y<l f(x% y= 0, otherwise Find (i) the value of c, (i) the marginal probability density functions (p.d.f.) of X and Y, (ii) the joint cumulative distribution function (c.d.f.) of X and Y, (iv) cov(X, Y) (covariance of X and Y) and (v) the correlation between X and Y, and comment upon the type of relationship. 2+4+4+4+6=20 DJSM-U-STSC/58 3 [ P.T.O. (b) Let X,, Xo, ..., X, be a random sample from Poisson distribution with mean A. “4 r>0 e”, Suppose prior distribution of A over (0, ~) is given by m(A)= i 0 , otherwise Find the Bayes estimator of A under quadratic error loss function. Is the prior distribution conjugate? Justify your answer. 10 (c) A random sample of size 10 from a continuous distribution over [0, 1] is as below : 0-73, 0:25, 0:48, 0-68, 0:79, 0-89, 0-61, 0:96, 0:54, 0:41 Check whether the data were randomly drawn from uniform distribution over [0, 1], using Kolmogorov-Smirnov goodness of fit test. (The critical value for a=0-05 is 0-409) 10 4. (a) (j) Let X and Y be discrete random variables with the probability mass function given as xX 1 2 3 1 1 1 2 | 6 2B 1 1 Y 3 6 0 A 1 4 0 3 0 Check whether X and Y are independent random variables. (i) Let X follow binomial distribution with parameters n and p. Find the characteristic function (c.f.) of X. Using the c.f., find E(X) and V(X). 5+5=10 (b) () Let {X, |n2=0} be a Markov Chain (MC) with state space {0, 1, 2}. The transition probability matrix (t.p.m.) is given as 012 ojo 10 P=1j30 4 2}0 1 0 Check whether the given MC is irreducible. Find the period of each state. Identify the recurrent states. 14+2+2=5 (ii) Let X be the number of ice blocks in a drink with probability mass function (p.m.f.) x 3 4 5 6 P(X = x) 0-4 0:3 0-1 0-2 For a sample of 49 drinks, find P(X <4-5). (Refer the tables enclosed) 5 DJSM-U-STSC/58 4 STANDARD NORMAL DISTRIBUTION Table Values Represent AREA to the LEFT of the Z score Zz 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.50000 0.50399 0.50798 0.51197 0.51595 0.51994 0.52392 0.52790 0.53188 0.53586 0.1 0.53983 0.54380 0.54776 0.55172 0.55567 0.55962 0.56356 0.56749 0.57142 0.57535 0.2 0.57926 0.58317 0.58706 0.59095 0.59483 0.59871 0.60257 0.60642 0.61026 0.61409 0.3 0.61791 0.62172 0.62552 0.62930 0.63307 0.63683 0.64058 0.64431 0.64803 0.65173 0.4 0.65542 0.65910 0.66276 0.66640 0.67003 0.67364 0.67724 0.68082 0.68439 0.68793 0.5 0.69146 0.69497 0.69847 0.70194 0.70540 0.70884 0.71226 0.71566 0.71904 0.72240 0.6 0.72575 0.72907 0.73237 0.73565 0.73891 0.74215 0.74537 0.74857 0.75175 0.75490 0.7 0.75804 0.76115 0.76424 0.76730 0.77035 0.77337 0.77637 0.77935 0.78230 0.78524 0.8 0.78814 0.79103 0.79389 0.79673 0.79955 0.80234 0.80511 0.80785 0.81057 0.81327 0.9 0.81594 0.81859 0.82121 0.82381 0.82639 0.82894 0.83147 0.83398 0.83646 0.83891 1.0 0.84134 0.84375 0.84614 0.84849 0.85083 0.85314 0.85543 0.85769 0.85993 0.86214 1.1 0.86433 0.86650 0.86864 0.87076 0.87286 0.87493 0.87698 0.87900 0.88100 0.88298 1.2 0.88493 0.88686 0.88877 0.89065 0.89251 0.89435 0.89617 0.89796 0.89973 0.90147 1.3 0.90320 0.90490 0.90658 0.90824 0.90988 0.91149 0.91309 0.91466 0.91621 0.91774 1.4 0.91924 0.92073 0.92220 0.92364 0.92507 0.92647 0.92785 0.92922 0.93056 0.93189 1.5 0.93319 0.93448 0.93574 0.93699 0.93822 0.93943 0.94062 0.94179 0.94295 0.94408 1.6 0.94520 0.94630 0.94738 0.94845 0.94950 0.95053 0.95154 0.95254 0.95352 0.95449 1.7 0.95543 0.95637 0.95728 0.95818 0.95907 0.95994 0.96080 0.96164 0.96246 0.96327 1.8 0.96407 0.96485 0.96562 0.96638 0.96712 0.96784 0.96856 0.96926 0.96995 0.97062 1.9 0.97128 0.97193 0.97257 0.97320 0.97381 0.97441 0.97500 0.97558 0.97615 0.97670 2.0 0.97725 0.97778 0.97831 0.97882 0.97932 0.97982 0.98030 0.98077 0.98124 0.98169 2.1 0.98214 0.98257 0.98300 0.98341 0.98382 0.98422 0.98461 0.98500 0.98537 0.98574 2.2 0.98610 0.98645 0.98679 0.98713 0.98745 0.98778 0.98809 0.98840 0.98870 0.98899 2.3 0.98928 0.98956 0.98983 0.99010 0.99036 0.99061 0.99086 0.99111 0.99134 0.99158 2.4 0.99180 0.99202 0.99224 0.99245 0.99266 0.99286 0.99305 0.99324 0.99343 0.99361 2.5 0.99379 0.99396 0.99413 0.99430 0.99446 0.99461 0.99477 0.99492 0.99506 0.99520 2.6 0.99534 0.99547 0.99560 0.99573 0.99585 0.99598 0.99609 0.99621 0.99632 0.99643 2.7 0.99653 0.99664 0.99674 0.99683 0.99693 0.99702 0.99711 0.99720 0.99728 0.99736 2.8 0.99744 0.99752 0.99760 0.99767 0.99774 0.99781 0.99788 0.99795 0.99801 0.99807 2.9 0.99813 0.99819 0.99825 0.99831 0.99836 0.99841 0.99846 0.99851 0.99856 0.99861 3.0 0.99865 0.99869 0.99874 0.99878 0.99882 0.99886 0.99889 0.99893 0.99896 0.99900 3.1 0.99903 0.99906 0.99910 0.99913 0.99916 0.99918 0.99921 0.99924 0.99926 0.99929 3.2 0.99931 0.99934 0.99936 0.99938 0.99940 0.99942 0.99944 0.99946 0.99948 0.99950 3.3 0.99952 0.99953 0.99955 0.99957 0.99958 0.99960 0.99961 0.99962 0.99964 0.99965 3.4 0.99966 0.99968 0.99969 0.99970 0.99971 0.99972 0.99973 0.99974 0.99975 0.99976 3.5 0.99977 0.99978 0.99978 0.99979 0.99980 0.99981 0.99981 0.99982 0.99983 0.99983 3.6 0.99984 0.99985 0.99985 0.99986 0.99986 0.99987 0.99987 0.99988 0.99988 0.99989 3.7 0.99989 0.99990 0.99990 0.99990 0.99991 0.99991 0.99992 0.99992 0.99992 0.99992 3.8 0.99993 0.99993 0.99993 0.99994 0.99994 0.99994 0.99994 0.99995 0.99995 0.99995 3.9 0.99995 0.99995 0.99996 0.99996 0.99996 0.99996 0.99996 0.99996 0.99997 0.99997 DJSM-U-STSC/58 5 [ P.T.O. STANDARD NORMAL DISTRIBUTION Table Values Represent AREA to the LEFT of the Z score Zz 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 -3.9 [0.00005 0.00005 0.00004 0.00004 0.00004 0.00004 0.00004 0.00004 0.00003 0.00003 -3.8 | 0.00007 0.00007 0.00007 0.00006 0.00006 0.00006 0.00006 0.00005 0.00005 0.00005 -3.7 | 0.00011 0.00010 0.00010 0.00010 0.00009 0.00009 0.00008 0.00008 0.00008 0.00008 -3.6 | 0.00016 0.00015 0.00015 0.00014 0.00014 0.00013 0.00013 0.00012 0.00012 0.00011 -3.5 | 0.00023 0.00022 0.00022 0.00021 0.00020 0.00019 0.00019 0.00018 0.00017 0.00017 -3.4 | 0.00034 0.00032 0.00031 0.00030 0.00029 0.00028 0.00027 0.00026 0.00025 0.00024 -3.3 | 0.00048 0.00047 0.00045 0.00043 0.00042 0.00040 0.00039 0.00038 0.00036 0.00035 -3.2 | 0.00069 0.00066 0.00064 0.00062 0.00060 0.00058 0.00056 0.00054 0.00052 0.00050 -3.1 | 0.00097 0.00094 0.00090 0.00087 0.00084 0.00082 0.00079 0.00076 0.00074 0.00071 -3.0 [0.00135 0.00131 0.00126 0.00122 0.00118 0.00114 0.00111 0.00107 0.00104 0.00100 -2.9 |0.00187 0.00181 0.00175 0.00169 0.00164 0.00159 0.00154 0.00149 0.00144 0.00139 -2.8 | 0.00256 0.00248 0.00240 0.00233 0.00226 0.00219 0.00212 0.00205 0.00199 0.00193 -2.7 | 0.00347 0.00336 0.00326 0.00317 0.00307 0.00298 0.00289 0.00280 0.00272 0.00264 -2.6 |0.00466 0.00453 0.00440 0.00427 0.00415 0.00402 0.00391 0.00379 0.00368 0.00357 -2.5 | 0.00621 0.00604 0.00587 0.00570 0.00554 0.00539 0.00523 0.00508 0.00494 0.00480 -2.4 |0.00820 0.00798 0.00776 0.00755 0.00734 0.00714 0.00695 0.00676 0.00657 0.00639 -2.3 {0.01072 0.01044 0.01017 0.00990 0.00964 0.00939 0.00914 0.00889 0.00866 0.00842 -2.2 |0.01390 0.01355 0.01321 0.01287 0.01255 0.01222 0.01191 0.01160 0.01130 0.01101 -2.1 [0.01786 0.01743 0.01700 0.01659 0.01618 0.01578 0.01539 0.01500 0.01463 0.01426 -2.0 |0.02275 0.02222 0.02169 0.02118 0.02068 0.02018 0.01970 0.01923 0.01876 0.01831 -1.9 |0.02872 0.02807 0.02743 0.02680 0.02619 0.02559 0.02500 0.02442 0.02385 0.02330 -1.8 |0.03593 0.03515 0.03438 0.03362 0.03288 0.03216 0.03144 0.03074 0.03005 0.02938 -1.7 |0.04457 0.04363 0.04272 0.04182 0.04093 0.04006 0.03920 0.03836 0.03754 0.03673 -1.6 | 0.05480 0.05370 0.05262 0.05155 0.05050 0.04947 0.04846 0.04746 0.04648 0.04551 -1.5 [0.06681 0.06552 0.06426 0.06301 0.06178 0.06057 0.05938 0.05821 0.05705 0.05592 -1.4 |0.08076 0.07927 0.07780 0.07636 0.07493 0.07353 0.07215 0.07078 0.06944 0.06811 -1.3 | 0.09680 0.09510 0.09342 0.09176 0.09012 0.08851 0.08691 0.08534 0.08379 0.08226 -1.2 |0.11507 0.11314 0.11123 0.10935 0.10749 0.10565 0.10383 0.10204 0.10027 0.09853 -1.1 |0.13567 0.13350 0.13136 0.12924 0.12714 0.12507 0.12302 0.12100 0.11900 0.11702 -1.0 |0.15866 0.15625 0.15386 0.15151 0.14917 0.14686 0.14457 0.14231 0.14007 0.13786 -0.9 |0.18406 0.18141 0.17879 0.17619 0.17361 0.17106 0.16853 0.16602 0.16354 0.16109 0.8 [0.21186 0.20897 0.20611 0.20327 0.20045 0.19766 0.19489 0.19215 0.18943 0.18673 0.7 =| 0.24196 0.23885 0.23576 0.23270 0.22965 0.22663 0.22363 0.22065 0.21770 0.21476 0.6 {0.27425 0.27093 0.26763 0.26435 0.26109 0.25785 0.25463 0.25143 0.24825 0.24510 0.5 | 0.30854 0.30503 0.30153 0.29806 0.29460 0.29116 0.28774 0.28434 0.28096 0.27760 0.4 | 0.34458 0.34090 0.33724 0.33360 0.32997 0.32636 0.32276 0.31918 0.31561 0.31207 0.3 | 0.38209 0.37828 0.37448 0.37070 0.36693 0.36317 0.35942 0.35569 0.35197 0.34827 0.2 | 0.42074 0.41683 0.41294 0.40905 0.40517 0.40129 0.39743 0.39358 0.38974 0.38591 0.1 | 0.46017 0.45620 0.45224 0.44828 0.44433 0.44038 0.43644 0.43251 0.42858 0.42465 -0.0 [0.50000 0.49601 0.49202 0.48803 0.48405 0.48006 0.47608 0.47210 0.46812 0.46414 DJSM-U-STSC/58 6 (c) Define one-parameter exponential family of distributions and verify whether the following distributions belong to the one-parameter exponential family of distributions : ) , x= 1 Dy ass. 0 22-21) ‌ “ 0 ; otherwise ) 4 -(-0/2. xs {> () = 12 0 ; otherwise 1 if 6 < × <6+1 1] 219) = (0 Fx (18) fo ; otherwise (v) > () = | 42706 650 0 ; otherwise 5+5+5+5=20 SECTION—B 5. (a) Compute the Mahalanobis distance of a point (3, 5) from the sample mean 4 _ , „ [2.5 3.25 vector with the sample variance matrix 9 6-4 3.25 4.7 8 () For the purpose of simple linear regression, the following data set (X, Y) is taken : x | 8 | | 6 | | 9 | 10 12 4 10 5 18 7 10 Study the sensitivity of the outlier (9, 18) in the given data and interpret. 8 ( Asample of 50 students is to be drawn from a population of 500 students from two colleges A and B. The means and standard deviations of their scores are given below : College Total number of Mean Standard students deviation A 300 40 10 | | 200 50 40 How would you draw the samples using proportional allocation? Hence, obtain the variance of estimate of the population mean and compare its efficiency with simple random sampling without replacement. 8 DJSM-U-STSC/58 7 [ P.T.O. (d) The following data were obtained from an experiment involving four tropical feeds A, B, Cand D tried on 20 chicks. All the 20 chicks were treated alike in all respects except the feeding treatments : Feed Gain in weight (in gm) A 55 49 42 21 52 B 61 112 30 89 63 c 42 97 81 95 92 D 169 137 169 85 154 Test whether all the feeds are equally effective at a = 0-05 level of significance. (Refer the table enclosed at the end) 8 (e) Suppose the random variables X,, Xj, X3 have covariance matrix 2 0-5 =-1 Z=|0-5 1 0-5 -1 0-5 2 Find the first principal component vector and its variance. 8 2 6. (a) Let X,, X, X3 be uncorrelated random variables having common variance o and E(X,)=8, +205, E(X)=20, +0, and E(X3)= 20, +465. (J Obtain the normal equations and the least square estimates 6; and 65. (ii) Find V(@6,), V6.) and cov(@,, 65). 15 (b) (i) Let X be a vector random variable with the following covariance matrix : 8 1 = 8 5 1 ilo nie nic De Y, =X, +2X_ +X, and Y, =3Xy-X, For what value of @ are Y, and Y, uncorrelated? Also, find the variance of Y,. 5 DJSM-U-STSC/58 8 (c) (ij) X and Y are random vectors with cov(X)=%,,, cov(Y)=Xo. and cov(X, Y)= = such that 1 0-4 0-5 0-6 z=(P ane 1 0-3 0-4 Yo, Eq) |0-5 0-3 1 0-2 0-6 0-4 0-2 1 Ee with 2, ? = 1-:0681 -—0-2229 1 1-0417 -0-2083 1 and X55 = -0-2229 1-0681 -0-2083 1-0417 first canonical correlation. | Find the Suppose X is a p-variate multivariate normal random vector and Y is also a p-variate multivariate normal random vector. X and Y are independent with E(X) =u, and E(Y) =, and with V(X) = V(Y) =. If n, observations are obtained from X and ny observations are obtained from Y, derive the test statistic for testing Ho:W, =Wy against H,:4, #4. Mention the distribution of the test statistic under null hypothesis. (a) Find out the missing entries denoted by * in the following ANOVA table of a Latin square design : Source of af Sum of Mean sum of Bist variation squares squares (SS) Rows 3 259-54 * * Columns * * * * Treatments * 1372-12 * * Error 6 * 26:06 Total * 1943-07 (b) (¢) Stating the hypotheses, give your conclusions at « =0-05 level of significance. Also, calculate the relative efficiency of the above design (LSD) with respect to RBD and CRD. (Refer the table enclosed at the end) (i) Show that systematic sampling is more precise than simple random sampling if the variance within the systematic samples is more than the total variation in the population. (ii) When do we make use of ratio estimator? Also, state the conditions under which the ratio estimate is optimum. Differentiate between partial and complete confounding, and write the ANOVA of a partially confounded 2°-experiment. 15 15 10 10 DJSM-U-STSC/58 9 [ P.T.O. 8. (a) The following is a sample of (X, Y, Z) of size 5 : (2, 6, 3), (3, 6, 4), (4, 8, 7), (5, 10, 9), (6, 10, 7) Find the partial correlation r,,., and multiple correlation coefficient Ry. ,.- (b) The surface finish of products produced in a machine shop is suspected to be affected by a factor ‘Operator’ and another factor ‘Shift’. The data of this experiment with two replications in different treatment combinations are given below : Shift (B) 1 2 Operator F 65 20 (4) 70 40 30 50 2 35 40 Test whether Operators, Shifts or their joint effects are significantly different at a =0-05. (Refer the table enclosed at the end) (c) (j) In which situations is cluster sampling preferred? Obtain the variance of sample mean of simple random sample (without replacement) of n clusters from a population of N clusters each having M elements. (i) What are non-sampling errors? Write in brief about the measures to control these errors. 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