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 .

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)?

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

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 )

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].

3. For each of the following pairs of random variables X and Y , the conditional distribution of Ygiven {X = x} is a named distribution, with parameters that may depend on x. For each pair:1i. Write down the conditional distribution of Y given {X = x}, including parameter(s).(2 marks each)ii. Find E (Y | X). (1 mark each)iii. Find E(Y ). (1 mark each)iv. Find Cov(X, Y ). (2 marks each)You do not need to find the marginal distributions of X and Y , which might not be nameddistributions.(a) Choose a number between 10 and 29, uniformly at random, and call that number X.Take a standard deck of 52 cards and lay them out in a row, face up. Using a pen, puta checkmark on the first X cards. Then turn the cards face down, shuffle throughly, anddraw 5 cards. Let Y be the number of marked cards you draw. [6 marks](b) Seta’s junior rugby team has twenty members. The team’s coach and parents decided todraw prize(s) to motivate the children to always do their best and to attend all activitieseach week. Each player (child) is assigned an ID number between 1 and 20 for every prizedraw. The team, the coaches and parents get together every Saturday afternoon for theprize draw. Assume that there is exactly one prize drawn at random each week. Seta (IDnumber 7) is a very smart and competitive player. He decided to roll a fair four-sided dieonce and whatever number is rolled, that’s the number of prize/s he wishes to get. LetX be the number that Seta has rolled (that is, the number of prize/s Seta wishes to get)and let Y be the number of weeks (or Saturdays) that Seta wins no prizes before he winsa prize for the Xth time.[6 marks](c) Roll a standard 4-sided die (with sides labelled 1, 2, 3, 4) and let X be the number rolled.Then take a fair 10-sided die (with sides labelled 1, . . . , 10) and roll it repeatedly until youroll a number strictly larger than X. Let Y be the number of times you roll the 10-sideddie, not including the last roll (i.e., the number of times you roll a number less than orequal to X)