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)?

1. Let X and Y be two random variables with joint pmf as follows.yfX,Y (x, y) -1 1 2x-2 0.15 0.10 0.051 0.25 0.25 0.20(a) Let event A = {{X is even} ∩ {Y is odd}}. Find PA
. [2 marks](b) Find the marginal pmf’s fX (x) and fY (y). [3 marks](c) Find E(XY ). [2 marks](d) Find E(X + Y ). [2 marks](e) Are X and Y independent? Justify your answer. [1 mark](f) Let event A = {{X is even} ∩ {Y is odd}}. Compute the conditional probability massfunction fY |A(y) for Y given A occurs. [3 marks](g) Find the conditional pmf fX|Y (x|2). [2 marks](h) Find E[X|Y = 2]. [2 marks](i) Find Cov(X, Y ). [2 marks](j) Find Var(X + Y). [4 marks](k) Compute the correlation ρX,Y .

Let X and Y be two random variables with joint probability mass function:P(X=i,Y=j) = (i+j)/36, for i,j=1,2,3,4What is the marginal probability mass function of X?Review LaterP(X=i) = 5(i+1)/36, for i=1,2,3,4P(X=i) = (5+i)/36, for i=1,2,3,4P(X=i) = 5/12, for i=1,2,3,4P(X=i) = (5i+1)/36, for i=1,2,3,4

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.Y1 2 3X -1 0.1 0.2 0.31 0.2 0.1 0.1Table 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

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

Quiz question. Random variables X and Y have joint p.m.f. shown in the following table: y fX,Y (x, y) 0 1 x 1 2 0.2 0.8 0.3 1 (a) Fill in the missing values. (b) Find P(X + Y ≤ 2). (c) Find E(XY )