Here is a new proof problem:

We have seen the law of total probability and 
the law of total expectation and we like the movie 
Total Recall, so lets prove the law of 
total variance. It  goes as follows:
Var(X) = E(var(X|Y)) + var(E(X|Y)).

(a) First show that  E(E(X|Y)) = E(X)

(b) Explain why the above is true.

(c) Use the above fact and show that Var(X) = E(var(X|Y)) + var(E(X|Y))

Solution:

    var(X) = E(X^2) - E(X)^2
    = E(E(X^2|Y)) - E(E(X|Y))^2
    = E(var(X|Y)) + E(E(X|Y)^2) - E(E(X|Y))^2
    = E(var(X|Y)) + var(E(X|Y)).


The following 3 are from CPs and PSets:

cp4f

Problem 4. Extra Problem.
(a) Exhibit three nonisomorphic, connected graphs with five vertices and four edges.
(b) Argue that every connected graph with five vertices and four 
    edges is isomomorphic to one of the three in part (a).

cp14w

Problem 5. The covariance, Cov [X, Y ], of two random variables, X and Y , 
is defined to be E [XY ]- E [X] E [Y ]. 
Note that if two random variables are independent, then their covariance is zero.

(a) Give an example to show that having Cov [X, Y ] = 0 
    does not necessarily mean that X and Y are independent.

(b) Let X1, . . . ,Xn be random variables. Prove that
    Var [X1 + · · · + Xn] =  Sum i=1..n Var [Xi] + 2 Sum i<j Cov [Xi,Xj ]


PS9

Problem 1. Professor Plum, Mr. Green, and Miss Scarlet are 
all plotting to shoot Colonel Mustard. If one of these three 
has both an opportunity and the revolver, then that person 
shoots Colonel Mustard. Otherwise, Colonel Mustard escapes. 
Exactly one of the three has an opportunity with the 
following probabilities:

Pr {Plum has opportunity} = 1/6
Pr {Green has opportunity} = 2/6
Pr {Scarlet has opportunity} = 3/6

Exactly one has the revolver with the following probabilities, 
regardless of who has an opportuntity:

Pr {Plum has revolver} = 4/8
Pr {Green has revolver} = 3/8
Pr {Scarlet has revolver} = 1/8

(a) Draw a tree diagram for this problem. 
    Indicate edge and outcome probabilities.
(b) What is the probability that Colonel Mustard is shot?
(c) What is the probability that Colonel Mustard is shot, 
    given that Miss Scarlet does not have the revolver?
(d) What is the probability that Mr. Green had an opportunity, 
    given that Colonel Mustard was shot?