Question 1: Recall, from your lecture slides, that if X1 is a chi-square random variable with v1 degrees of freedom, and X2 is a chi-square random variable with v2 degrees of freedom, and X1 and X2 are independent, then Z = X1 + X2 follows a chi-square distribution with (v1 + v2) degrees of freedom. Suppose that X3 is a chi-square random variable with v3 degrees of freedom, and it is independent of both X1 and X2. Then the random variable, Y = (X1 + X2 - X3) does not follow a chi-square distribution!
(a) What is the mean of the random variable, Y? What is the variance of Y?
(b) Suppose that we have n1 random sample values from the distribution of X1, n2 random sample observations from the distribution of X2, and n3 random sample observations from the distribution of X3. Suggest an unbiased estimator for the mean of Y that you have stated in part (a)
(c) Suggest an unbiased estimator for the variance of Y that you have stated in part (a).
(d) Are the estimators that you have suggested in parts (b) and (c) consistent? Why or why not?