(30 points) Suppose Company 1's stock return 𝑋𝑋 is a random variable and takes three possiblevalues: {-0.1, 0.1, 0.2}. And Company 2's stock return 𝑌𝑌 is a random variable and takes twopossible values: {-0.3, 0.4}. The joint probability distribution 𝑓𝑓(𝑋𝑋, 𝑌𝑌) is given as follows:𝑓𝑓(−0.1, −0.3) = 0.1, 𝑓𝑓(0.1, −0.3) = 0.2, 𝑓𝑓(0.2, −0.3) = 0.2,𝑓𝑓(−0.1,0.4) = 0.2, 𝑓𝑓(0.1,0.4) = 0.2, 𝑓𝑓(0.2,0.4) = 0.1.Please calculate the following:(a) Marginal distributions: 𝑓𝑓𝑋𝑋(𝑥𝑥) and 𝑓𝑓𝑌𝑌(𝑦𝑦). (4 points)(b) Mean: 𝐸𝐸(𝑋𝑋) and 𝐸𝐸(𝑌𝑌). (4 points)(c) Variance: 𝑉𝑉𝑎𝑎𝑟𝑟(𝑋𝑋) and 𝑉𝑉𝑎𝑎𝑟𝑟(𝑌𝑌). (4 points)(d) Covariance: 𝐶𝐶𝐶𝐶𝐶𝐶(𝑋𝑋, 𝑌𝑌). (2 points)(e) Conditional expectations: 𝐸𝐸(𝑋𝑋|𝑌𝑌 = −0.3) and 𝐸𝐸(𝑋𝑋|𝑌𝑌 = 0.4). (8 points)(f) Conditional variances: 𝑉𝑉𝑎𝑎𝑟𝑟(𝑋𝑋|𝑌𝑌 = −0.3) and 𝑉𝑉𝑎𝑎𝑟𝑟(𝑋𝑋|𝑌𝑌 = 0.4). (8 points)

Sure, let's solve this step by step:

(a) Marginal distributions: 𝑓𝑋(𝑥) and 𝑓𝑌(𝑦). The marginal distribution of X is the sum of the joint probabilities for each value of X, regardless of Y. Similarly, the marginal distribution of Y is the sum of the joint probabilities for each value of Y, regardless of X.

For X: fX(-0.1) = f(-0.1, -0.3) + f(-0.1, 0.4) = 0.1 + 0.2 = 0.3 fX(0.1) = f(0.1, -0.3) + f(0.1, 0.4) = 0.2 + 0.2 = 0.4 fX(0.2) = f(0.2, -0.3) + f(0.2, 0.4) = 0.2 + 0.1 = 0.3

For Y: fY(-0.3) = f(-0.1, -0.3) + f(0.1, -0.3) + f(0.2, -0.3) = 0.1 + 0.2 + 0.2 = 0.5 fY(0.4) = f(-0.1, 0.4) + f(0.1, 0.4) + f(0.2, 0.4) = 0.2 + 0.2 + 0.1 = 0.5

(b) Mean: E(X) and E(Y). The mean (expected value) of a random variable is the sum of the product of each value and its probability.

For X: E(X) = -0.1fX(-0.1) + 0.1fX(0.1) + 0.2fX(0.2) = -0.10.3 + 0.10.4 + 0.20.3 = 0.05

For Y: E(Y) = -0.3fY(-0.3) + 0.4fY(0.4) = -0.30.5 + 0.40.5 = 0.05

(c) Variance: Var(X) and Var(Y). The variance of a random variable is the expected value of the squared deviation from the mean.

For X: Var(X) = E[(X - E(X))^2] = (-0.1 - 0.05)^2fX(-0.1) + (0.1 - 0.05)^2fX(0.1) + (0.2 - 0.05)^2*fX(0.2) = 0.0225

For Y: Var(Y) = E[(Y - E(Y))^2] = (-0.3 - 0.05)^2fY(-0.3) + (0.4 - 0.05)^2fY(0.4) = 0.2025

(d) Covariance: Cov(X, Y). The covariance between two random variables is the expected value of the product of their deviations from the mean.

Cov(X, Y) = E[(X - E(X))(Y - E(Y))] = (-0.1 - 0.05)*(-0.3 - 0.05)f(-0.1, -0.3) + ... + (0.2 - 0.05)(0.4 - 0.05)*f(0.2, 0.4) = -0.0075

(e) Conditional expectations: E(X|Y = -0.3) and E(X|Y = 0.4). The conditional expectation of X given Y is the expected value of X when Y is known to be a certain value.

E(X|Y = -0.3) = -0.1f(-0.1, -0.3)/fY(-0.3) + 0.1f(0.1, -0.3)/fY(-0.3) + 0.2f(0.2, -0.3)/fY(-0.3) = 0.1 E(X|Y = 0.4) = -0.1f(-0.1, 0.4)/fY(0.4) + 0.1f(0.1, 0.4)/fY(0.4) + 0.2f(0.2, 0.4)/fY(0.4) = 0

(f) Conditional variances: Var(X|Y = -0.3) and Var(X|Y = 0.4). The conditional variance of X given Y is the variance of X when Y is known to be a certain value.

Var(X|Y = -0.3) = E[(X - E(X|Y = -0.3))^2|Y = -0.3] = (-0.1 - 0.1)^2f(-0.1, -0.3)/fY(-0.3) + (0.1 - 0.1)^2f(0.1, -0.3)/fY(-0.3) + (0.2 - 0.1)^2f(0.2, -0.3)/fY(-0.3) = 0.01 Var(X|Y = 0.4) = E[(X - E(X|Y = 0.4))^2|Y = 0.4] = (-0.1 - 0)^2f(-0.1, 0.4)/fY(0.4) + (0.1 - 0)^2f(0.1, 0.4)/fY(0.4) + (0.2 - 0)^2f(0.2, 0.4)/fY(0.4) = 0.02