Sampling Distributions — DU B.A. (Prog.) with Statistics as Major Discipline Previous Year Question Papers

DSC: Sampling Distributions (NEP) B.Sc. (Hons.) Statistics Attempt six questions in all by selecting three questions from each section. 1. (a) State Chebychev's Inequality. If X has the distribution with p.d.f. /(I) = e-.r,x 2 0 , use Chebychev's inequality to obtain a lower bound to probability of the inequality -I S X < 3 , and compare it with the actual probability. (b) If X, can have only two values with equal probabilities z.a and - 7.a, show that 1 WLLN can be applied to the independent variables ,Y, , .X2 ,..., if a < - . 2 2. (a) If X],JY2„ . . ,X„ are i.i.d. random variables with mean A and variance cr2 (finite). (b) Let .Y, , JF2 ,..., X„ be a random sample of size n from a population with p.d.f. •/(„0<x<I.Showthatr=#,y2=#,...,y„_,=¥;:landr,=Xt„,are independently distributed and identify their distributions. 3. (a) Define convergence in law, convergence in probability and convergence with probability one. State the relationship between convergence in probability and convergence with probability one and prove it. (b) For the exponential distribution, /(x) = e-.t,x 2 0 , find the p.d.f. of the sample range W in a random sample of size n and show that E[J7] = 1+i+i+...+i 2 3 „-1 4. (a) What is meant by a statistical hypothesis? What are the two types of errors of decision that arise in testing a hypothesis? Briefly explain how a statistical hypothesis is tested. (b) Explain the following terms: (i) P-value and I.evel of significance (ii) Critical region and significant value (iii) Sampling distribution of a statistic (iv) Test statistic for the test of significance for difference of proportions 5. (a) Find the mean deviation about mean for t-distribution with n d.f. (b) If F ~ F77„ and lim 77 + co then prove that "F follows chi-square distribution with ,7' d.f„ (c) What is contingency table? Describe how the chi-square distribution may be used to test whether the two attributes in an mxn contingency table are independent. 6. (a) If X and Y are independent chi-square variates with m and n d.f. respectively, Show that U = + Y and V - my are independently distribured. Also, Find the distribution of V. (b) It is required to test that the mean of a normal population is zero against it is greater than zero. Show that the t-test for acceptance of hypothesis based on a random sample of size n reduces to Ex) - M (Ex), where a is the value of student's t at the desired level of significance a for (n-1) degrees of freedom. 7. (a) X is a Poisson variate with parameter 2 and x' is a chi-square variate with 2k d.f. Prove that for all positive integers, PIX ≤k-I]= PIX > 22] (b) Let X, and X, are independently distributed as 2'-variate with 2 d.f. Find the distribution of Z=- 8. (a) Suppose X,, X2,.... X, (n> 1) are independent variates each distributed as N(0,0'). Find the p.d.f. of W= *,. Why does not W follow the t- (b) Prove that if n, = n2, the median of F-distribution is at F = 1 and that the quartiles O, and O, satisfy the condition 903 =1.