If x1, x2, . . , xn are independent and identically distributed random variables having uniform distributions over (0, 1), find (a) e[max(x1, . . , xn)]; (b) e[min(x1, . . , xn)].

Denote by the maximum order statistic, with , and similarly denote by the minimum order statistic. Then the CDF for is



In order for there to be some that exceeds the value of , it must be true that exceeds the value of all the , so the above is equivalent to the joint probability



and since the are i.i.d., we have




where . We have



and so





Using similar reasoning, we can find the CDF for . We have