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 )

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

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 .

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 value of c. Find P(Y<X) and P(Y=X).

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].

The joint probabilities are given in the table. The value of P(Y<=2) is3/161/163/323/64