7. Let X1....,Xn lit} Uniform(0,1). Find the probability density function for V = Yn Y1, where Y1 =min{X1,...,Xn} and Yn =max{X1,...,Xn}. 8. A scale measures weight in pounds, that are distributed following a normal distribution with an un known variance. Suppose we estimate the variance to be 32 from n = 9 random sample of measure ments. Find a and I) such that ngYpgm=u%. Is there an ambiguity in choosing a. and b? What would be a reasonable decision to make? Note: Your answer will be a function of S. 9. (a) Let V N X2 with 1/ degrees of freedom. Use the moment generating function to nd E(V) and V(V). (b) We know (71 1)S2 /02 N X2 with n 1 degrees of freedom. Using the result from previous part to compute lE(S2) and V(S2). 10. A fair coin is tossed 20 times and all tosses are independent. (a) What is the true probability of obtaining exactly 10 heads? Compute and compare the probability when approximated by an appropriate normal distribution. (b) Compute approximate probability of obtaining more than 7 heads both with, and without continuity correction. Compare these approximations to the exact probability found using pmf of binomial distribution