3. (Statistics Review) (a) (b) Suppose a random variable a: follow a normal distribution of N [%, (%)2]. Let z = 43:2 41' + 1, what is the distribution of z? What is its mean, i.e. E(z)? [Hintz (12:62 + 2abx + b2 = ((1512 + b)2 and use the property of the normal distribution and Chi-square distribution] Suppose Xn = {371, m2, - ' ' , 93\"} is a sample with n observations drawn from the distribution of a: in (a). For each observation 35,-, you can compute the corresponding 2,- according to the relation in (a). With {21, 22, - - - , 2\"}, you can then compute the sample average in = (1 ) 221:1 2,- as the mean estimate of 2:. For a sample size of n = 5, draw a radom sample Xn and compute an absolute error of the mean estimate for z according to, el) = |ZU E(z)|. Discard your sample of X\" and redraw a new 5-observation sample for X\". Compute and denote your second round error estimate as 6?). Repeat the exercise for 100 times. You need to report your average error, 55 = (e?) + a?) + - - - + 353100)) / 100 (0) Repeat (b) for n = 10, 100, and n = 500, respectively. With respect to the Law of Large Numbers, what can you say