= 7.1.7. Let X1, X2,..., Xn denote a random sample from a distribution that is N(μ,0), 0 < 0 <∞o, where u is unknown. Let Y (X-X)2/n and let L[0,8(y)] = [0-6(y)]². If we consider decision functions of the form 8(y) = by, where b does not depend upon y, show that R(0, 8) = (0²/n²) [(n² - 1)6² -2n(n-1)b+n²]. Show that b = n/(n+1) yields a minimum risk decision function of this form. Note that nY/(n+1) is not an unbiased estimator of 0. With 8(y) ny/(n + 1) and 0<0<∞, determine max, R(0,8) if it exists. =

Consider a set of data x1, x2, n n i=1 ..., n taken from a population with mean µ. - Show that (x-μ)² = Σ(x₂ − x)² + n(x − µ)². i=1

2.Suppose that the return R (in dollars per share) of a stock has the uniform distribution on the interval [-3,7]. Suppose also, that each share of the stock costs $1.50. Let Y be the net return (total return minus cost) on an investment of 10 shares of the stocks. Compute E(Y).