Min and max of iid uniforms¶ let ,…,u2,…,un be i. i.d. uniform on (0,1)(0,1) . let )u(1) and (n) be the minimum and maximum of ,…,u2,…,un . a) find the joint density of )u(1) and (n) . b) find the density of )u(1) . c) fix x∈(0,1)x∈(0,1) and find the conditional density of (n) given )=xu(1)=x . d) for fixed x∈(0,1)x∈(0,1) , let ,…−1x1,x2,…,xn−1 be −1n−1 i. i.d. uniform (x,1)(x,1) random variables. find the density of =max{,…−1}m=max{x1,x2,…,xn−1} and compare it to your answer to part c. e) the random variable =−)rn=u(n)−u(1) is called the range of the sample ,…,u2,…,un . find (rn) .