Can someone solve this?

Exercise 1. Let (2, F, P) be a probability space. Define A E F an atom of Q if A # @ and 3 and A are the only elements in F which are subsets of A. Show that distinct atoms are disjoint. Exercise 2. Let {Xn}nen a sequence of i.id.r.v. with E[X] = m + 0. Denoting by Sn = E1=1 XJ, prove that a.s. only a finite number of events [Sn E I ] happen if I C R is a bounded interval. Exercise 3. We throw a fair die. If we get an even result, we pick a random number in the interval [0, 1] of R, otherwise we pick a random number in the interval [0, 2] . Define X the number we have found. Compute the characteristic function px. Exercise 4. Let X be a standard normal random variable. Find the vari- ance of Y = ex. Exercise 5. Let X be a standard normal random variable. Let e a Rademacher random variable independent from X, i.e. P[E = -1] = P[( = 1] = ;. De- fine Y = EX. Show that Y is a normal random variable. Does the vector Z = (X, Y) have the bivariate normal distribution? Exercise 6. Let Y be a binomial random variable following the distribu- tion Binom(X,p), where 0 . Compute E[X|Y]