Let the discrete RV 𝑋~𝑈[−2,2] (Uniform dist.). Let 𝑌 = 𝑋2a) What values X and Y can take? Find pdf’s of both X and Y.b) Compute the joint pdf, 𝑓𝑋𝑌(𝑥𝑖, 𝑦𝑖)c) Compute the E(X) and E(Y)d) Compute the Cov(X,Y)e) Compute the 𝜌𝑋𝑌 = 𝐶𝑜𝑟(𝑋, 𝑌).f) Are X and Y independent? Prove it.

a) The random variable X can take any integer value between -2 and 2, inclusive. So, X = {-2, -1, 0, 1, 2}. The random variable Y = X^2, so Y can take values {0, 1, 4}. The pdf of X, f_X(x), is 1/5 for x in {-2, -1, 0, 1, 2} and 0 otherwise. The pdf of Y, f_Y(y), is 1/5 for y = 0, 2/5 for y = 1, and 2/5 for y = 4.

b) The joint pdf, f_XY(x, y), is the product of the marginal pdfs, f_X(x) and f_Y(y), if X and Y are independent. However, since Y is a function of X, they are not independent. Therefore, we cannot compute the joint pdf in this way.

c) The expected value E(X) is the sum of the product of each value of X and its probability, which is 0. The expected value E(Y) is the sum of the product of each value of Y and its probability, which is 1.6.

d) The covariance Cov(X, Y) is E(XY) - E(X)E(Y). Since E(X) = 0, Cov(X, Y) = E(XY) = 0.8.

e) The correlation ρ_XY = Cov(X, Y) / (σ_X σ_Y), where σ_X and σ_Y are the standard deviations of X and Y, respectively. Since Cov(X, Y) = 0.8, σ_X = sqrt(2), and σ_Y = sqrt(1.2), ρ_XY = 0.8 / (sqrt(2) * sqrt(1.2)) = 0.58.

f) X and Y are not independent because Y is a function of X. This is evident from the fact that the joint pdf cannot be expressed as the product of the marginal pdfs. Also, the correlation ρ_XY is not zero, which indicates dependence.