2. Suppose that Y = (Y1,...,Y) is a random sample from an N(, 1) distribution. Let U = Y and S = 1 Y. Show that U and S are jointly normally distributed. i=1 3. Suppose that Y = (Y,...,Y) is a random sample from a Bernoulli(p) distribution. Starting with the estimator p = Y, use the Rao-Blackwell theorem to find a better estimator. 4. Suppose that Y,.. .,Y, is a random sample from a population with density 2y fy(y)= = for 0 < y 0, where > 0. (a) Is the last order statistic Y(n) a sufficient statistic for 0? (b) Derive the density of Y(n). (c) Find an unbiased estimator of 0 that is a function of Y(n). (d) By considering the condition that a function of Y() is unbiased for 0, determine whether there is a better unbiased estimator for 0. 5. (a) Explain what is meant by minimum-variance unbiased estimator. (b) Let Y,..., be a random sample from the uniform distribution on the interval [0, 0], where > 0 is an unknown parameter. Define U = max{Y,...,Yn}. i. Show that U is a sufficient statistic for 0. ii. Show that = (n + 1)U/n is an unbiased estimator of 0. Find its variance. iii. Show that is a minimum-variance unbiased estimator for 0.