1. Let X1, X2, . . . , X8 be independent Exp(1)-distributed random variables with order statistic (X(1), X(2), . . . , X(8)). Find E(X(7) | X(5) = 10).
2. Let X1, X2, X3 be independent U (0, 1)-distributed random variables, and let X(1), X(2), X(3) be the order statistic. Prove the intuitively reasonable result that X(1) and X(3) are conditionally independent given X(2) and determine this (conditional) distribution. Remark. The problem thus is to determine the distribution of (X(1), X(3)) | X(2) = x.
3. The random variables X1, X2, and X3 are independent and Exp(1)- distributed. Compute the correlation coefficient ρX(1) ,X(3) .
4. Let X1 and X2 be independent, Exp(a)-distributed random variables.
a) Show that X(1) and X(2) - X(1) are independent, and determine their distributions.
b) Compute E(X(2) | X(1) = y) and E(X(1) | X(2) = x).
5. Let X1, X2, and X3 be independent, U (0, 1)-distributed random variables. Compute P (X(3) > 1/2 | X(1) =x).