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Mathematical Statistics

This document outlines the examination structure for the M.Sc. degree in Mathematics, specifically focusing on the elective course in Mathematical Statistics for the November 2024 semester. It includes various topics such as probability, distributions, and statistical theorems, with questions divided into parts A, B, and C, each requiring different types of responses. The examination consists of theoretical questions, problem-solving tasks, and proofs related to statistical concepts.

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61 views4 pages

Mathematical Statistics

This document outlines the examination structure for the M.Sc. degree in Mathematics, specifically focusing on the elective course in Mathematical Statistics for the November 2024 semester. It includes various topics such as probability, distributions, and statistical theorems, with questions divided into parts A, B, and C, each requiring different types of responses. The examination consists of theoretical questions, problem-solving tasks, and proofs related to statistical concepts.

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bscmathematicslectures
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S-3886 Sub. Code 23MMA2E2 (eee M.Sc. DEGREE EXAMINATION,-NOVEMBER 2024. Second Semester Mathematics Elective - MATHEMATICAL STATISTICS (CBCS - 2023 onwards) Time : 3 Hours Maximum : 75 Marks Part A (10 x 2 = 20) Answer all questions. Find the intersection cc, of the two sets c, and ¢ where ¢, = {(x, y)}00, then (x = is 70). prove that the random variable v= Or Derive the p.d.f of Dirichlet distribution. X,—2+a and the real function & is Suppose at g(X,) > (@)- continuous at ‘a’. Then prove th: Or If X,, converges to X in probability, then prove that X, converges to X in distribution. ustomers X that enter @ pose the number of ¢ 9:00 a.m and 10:00 a.m Sup store between the hours ribution wil le of the number of 0 a.m follows a poisson dist th parameter a. m samp. the store between 9:0 sults in the values Suppose 2 rando! customers that enter and 10:00 for 10 days res 979 15 190 18-7 2 12 Determine the maximum likelihood estimator of 6. Show that it is an unbiased estimator. Or Explain the zea mays data probelm. S-3886 16. ane 18. 19. 20. Part C (3 x 10 = 30) Answer any three questions. (a) Prove that for . any random _ variable, P[X=x]=Fy(x)-F,(-x) for all xeR, where Fy(x-)=lim,, Fy (2). (b) Let x be a continuous random variable with pdf fx(x) and support Sy. Let Y = g(X), where g(x) is a one-to-one differentiable function, on the support of X, Sy. Denote the inverse of g by x= g7'(y) and —1 let dx _ ale“). Then prove that the pdf of Y is Ly Ly given by fy(y)= fy (Ora , for ye Sy, where the Ly support of Y is the set Sy = {y= g(x) xe Sy}. Let (X,, X,) be a random vector such that the variance of X is finite, Then prove that (@) E[E(X,1X,)]= #(%,) (b) var[E(X, | X,)]< var(X,) Compute the measures of Skewness and Kurtosis of the poisson distribution with mean pw. State and prove the central limit theorem. A die was cast n=120 independent times and the following data resulted. Sportsup 1 2 3. 4 5 6 Frequency b 20 20 20 20 40-b If we use a chi-square test, the hypothesis that the die 0.025 significance level? 7 for what values of b would is unbiased be rejected at the eee 4 | g3eeq |

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