1. Let X be a random variable with values 1,2,3 equally likely, and let Y be independent with values 1,2,3,4,5,6 equally likely. Let Z=X+Y.
(a) Compute E[X|Z].
(b) Compute E[Z|X].
(c) Let W=XY. Compute E[W|Y].
(d) Compute E[Y|W].
2. Let 
be simple random walk. Show that 
is a martingale.
3. Let X be a Binomial(n,/1/2), and let Y be zero if X is even and 1 if X is odd. (a) Find E[X|Y] and E[Y|X].
4. Consider the experiment in which 4 independent coins are tossed. Let X be the number of heads, and let Y be the remainder when X is divided by 3. (That is, Y is 0,1 or 2).
(a) Describe explicitly the sigma-field sigma(Y).
(b) Find the distribution of Y.
(c) Find Z=E[X|Y].
(d) Find EZ
(e) Find E[Y|X].
5. Let X be a random variable with finite expectation, and let 
be sigma-fields so that 
.
Show that 
is a martignale.
6. Show that for any random variable X with finite second moment we have 
.