Random variable X may take value -1 or 1; Y may take value 1, 2 or 3.

The following Table shows the joint probability of random variables X and Y.

Y123X-10.10.20.210.10.30.1

Table 2: Joint probability distribution of X and Y. For example, the probability of X=-1 and Y=1 is 0.1.

Provide solution to all questions below.

(a) Marginal probability distribution of X

(b) The expected value, variance and standard deviation of X.

(c) Marginal probability distribution of Y

(d) The expected value, variance and standard deviation of Y.

(e) Conditional probability distribution of Y given X=-1.

(f)The conditional expected value, conditional variance and conditional standard deviation of Y given X=-1.

(g) Conditional probability distribution of Y given X=1.

(h) The conditional expected value, conditional variance and conditional standard deviation of Y given X=1.

(i) Verify the Law of Iterated Expectations E(Y)=E[E(Y|X)] by using your previous results.

(j) Covariance and correlation of X and Y