Let X and Y be the discrete random variables with the given pmf pX (xi)and pY (yi). Assume E[X2] < ∞, E[Y 2] < ∞. Variance and covariance are defined asV ar(X) = E[(X − E[X])2], Cov(X, Y ) = E[(X − E[X])(Y − E[Y ])]. Verify that• V ar(X) = E[X2] − E[X]2

Let X and Y be two discrete random variables with the given pmf pX (xi),i = 1, 2, . . . , n, and pY (yj ), j = 1, 2, . . . , m. Verify for following properties using thedefinition E[X] = ∑ni=1 xipX (xi).• (linearity) E[aX + bY + c] = aE[X] + bE[Y ] + c.• (monotonicity) If X ≤ Y , E[X] ≤ E[Y ].

Problem 1 (15 points). Suppose (X, Y ) are discrete random variables with joint probability massfunction given by the following table:x\y -2 0 2-2 3/16 1/16 3/160 1/16 0 1/162 3/16 1/16 3/16a. (6 points) Find the marginal pmf’s for X and Y and check if the two random variables areindependent.b. (5 points) Show that the covariance between X and Y is zero.c. (4 points) Is there any contradiction between the results in parts (a) and (b)?

Cov(X, Y ) = E[XY ] − E[X]E[Y ]

Context: The random variables X and Y have the joint PMF: px,y(x, y)=c*(x+y)^(2) if x belongs to {1,2,4} and y belongs to {1,3} and otherwise px,y(x,y) =0. Find the expectations E[XY].

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.