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

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).

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

X                                               Y Y=0 Y=1 Y=2X=0      0.32 0.03 0.01X=1 0.06 0.24 0.02X=2 0.02 0.03 0.27Find  P(X≤1, Y=1)

uppose Company 1’s simple return X is a random variable and takes three possiblevalues {−0.1, 0.1, 0.2}. Company 2’s simple return Y is a random variable and takes twopossible values {−0.3, 0.4}. The joint probability distribution f (X, Y ) is as follows:f (X = −0.1, Y = −0.3) = 0.2, f (X = 0.1, Y = −0.3) = 0.2, f (X = 0.2, Y = −0.3) = 0.1,f (X = −0.1, Y = 0.4) = 0.1, f (X = 0.1, Y = 0.4) = 0.2, f (X = 0.2, Y = 0.4) = 0.2Please calculate the following:(a). Marginal distributions: fX (x) and fY (y);(b). Mean: E[X] and E[Y ];(c). Variance: var(X) and var(Y );(d). Covariance: cov(X, Y );(e). Correlation: corr(X, Y )

The table gives a set of outcomes and their probabilities. Let A be the event "the outcome is less than or equal to 2". Let B be the event "the outcome is greater than or equal to 4". Find P(A or B).Outcome Probability1 0.152 0.043 0.044 0.665 0.11