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

4Q. The joint probability mass function of discrete random variables 𝑋 X and 𝑌 Y is given by: 𝑃 [ 𝑋 = 𝑥 , 𝑌 = 𝑦 ] = { 𝑘 ( 4 𝑥 + 𝑦 ) , 𝑥 = 1 , 2 , 3  and  𝑦 = 1 , 2 , 3 0 , otherwise P[X=x,Y=y]={ k(4x+y), 0, ​ x=1,2,3 and y=1,2,3 otherwise ​ Find the value of 𝑘 k. Then, calculate 𝑃 ( 𝑋 = 2  and  𝑌 = 3 ) P(X=2 and Y=3).

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

A discrete random variable Y has pmf (probability mass function) given by p=c/(y+4), y=-1,0,1,2, where c is a constant. Find the standard deviation of Y.