2. [15 points] Suppose that X,Y are discrete random variables with joint probability
         Y = −1 Y = 0 Y = 1
X = 0       0       ⅓       1
X = 1       ⅓    0       ⅓

(a) (3 pt) Compute the conditional distributions p(X|Y = y) for all possible values of y.
(b) (2 pt) Compute the conditional distributions p(Y|X = x) for all possible values of x.
(c) (3 pt) Compute E[X|Y = y] for all values of y.
(d) (2 pt) Plot E[X|Y = y] against y.
(e) (5 pt) Are X and Y dependent? If yes, explain why. If no, how do you think X and Y are related?

5. [25 points] Suppose that X is a random variable with values {0,1}, and p(X = 1) = θ.
In the following, express your answer in terms of θ.
(a) (3 pt) Define and calculate the expectation of X.
(b) (4 pt) Define and calculate the variance of X.
(c) (6 pt) Calculate the quantity φ(t) := E[e ͭ ͯ ] for a fixed t > 0
(d) (6 pt) Obtain an expression for the derivative φ' (t). Explain the relation of φ'(0) to
the result of (a).
(e) (6 pt) Obtain an expression for the second derivative φ''(t). Explain the relation of φ''(0) to the result of (b).

2. [15 points] Suppose that X,Y are discrete random variables with...