Help out with the following

Problem 1 Suppose a sequence of random variables is generated as follows. Let X] ~ uniform(0, 1), and X2 X] ~ uniform(X1, 1). In general, let Xxtil(X1, ..., Xx) ~ uniform (Xk, 1). (a) Find the joint density of (- log(1 - X1), - log(1 - X2)) by using bivariate trans- formations. Be sure to specify the support of this density. (b) Find the marginal density of - log(1 - X2). (c) Find the conditional density of X2 given (X1, X3). What about X2 given (X1, X3, X4, ..., X20 (d) Find EXe, k = 1,2,... by using the law of iterated conditional expectations. (Hint: you may find it easier to work with E(1 - Xk).) (e) Prove that Xx - 1 in probability as k -> co