denote the sample mean and denote the sample variance as defined earlier in the course. (a) For l s is " letD, - X, - X_ Find Coup, X). (b] Now assume in addition that Y1 X2 ..., *, are Lid. normal (#, "). What is the joint distribution of X, Dj. Dys . Dn-1? Explain why Do, isn't on the list. (c) True or false (justify your answer): The sample mean and sample variance of an Lid. normal sample are independent of each other. 6. Normal Sample Mean and Sample Variance, Part 2 (a) Let Rhave the chi-squared distribution with a degrees of freedom. What is the myf of R? (b) For R as in Part (a], suppose R - V+ Wwhere I and W are independent and Ihas the chi-squared distribution with w degrees of freedom. Can you identify the distribution of #? Justify your answer. (c) Let ALX2 .... *, be any sequence of random variables and letY - - E2-1-*, Let a be any constant. Prove the sum of squares decomposition (d) Now letX X2 ..., X, be Lid. normal with meany and variance = > 0. Let S' be the "sample variance" defined by 8 - -14 Find a constante such that of" has a chi-squared distribution. Provide the degrees of freedom. (Use Parts (b] and (c) as well as the result of the previous exercise ] By Ani Adhikari @ Copyright 2020. License: CC BY-NC-ND 4.0