Normal Sample Mean and Sample Variance, Part 2 a) Let have the chi-squared distribution with degrees of freedom. What is the mgf of ? b) For as in Part (a), suppose where and are independent and has the chi-squared distribution with degrees of freedom. Can you identify the distribution of ? Justify your answer. c) Let be any sequence of random variables and let . Let be any constant. Prove the sum of squares decomposition d) Now let be i.i.d. normal with mean and variance . Let be the "sample variance" defined by Find a constant such that has a chi-squared distribution. Provide the degrees of freedom. [Use Parts (b) and (c) as well as the result of the previous exercise.]