The joint probability function for the random variables X and Y, P(X,Y), is given in matrix below. Then, the mean value of X and Y are P(X,Y)= 1 1 [0.05 0.05 0 x 2 3 y 2 0.05 0.1 0.2 3 0.1 0.35 0.1 2.1, 2.1 2.1, 2.5 2.15, 2.45 2.1, 2.45 O O 2.2, 2.44 O
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- Suppose that the random variables Y1 and Y2 have joint probability distribution function. f(y1, y2) = 2, 0 ≤ y1 ≤ 1, 0 ≤ y2 ≤ 1, 0 ≤ y1 + y2 ≤ 1, 0, elsewhere (a) Use R to calculate P(Y1 ≥ 1⁄6 | Y2 ≤ 1⁄5). (Round your answer to four decimal places.) P(Y1 ≥ 1⁄6 | Y2 ≤ 1⁄5) = (b) Use R to calculate P(Y1 ≥ 1⁄6 | Y2 = 1⁄5). (Round your answer to four decimal places.) P(Y1 ≥ 1⁄6 | Y2 = 1⁄5) =If Y is a discrete random variable with possible values of 1, 2, and 4, and the probability mass function is given by P(Y = 1) = 0.2, P(Y = 2) = 0.5, and P(Y = 4) = 0.3, what is the variance of Y?If X and Y have the joint probability distributionf(−1, 0) = 0, f(−1, 1) = 1 4 , f(0, 0) = 16 , f(0, 1) = 0, f(1, 0) = 112 , and f(1, 1) = 12 , show that (a) cov(X, Y) = 0;(b) the two random variables are not independent.
- The joint probability function of two discrete random variables X and Y is given by Ax,y) = c(2x+y), where x and y can assume all integers such that 0< xIf Z is a discrete random variable with possible values of -1, 0, and 1, and the probability mass function is given by P(Z = -1) = 0.4, P(Z = 0) = 0.3, and P(Z = 1) = 0.3, what is the standard deviation of Z?(b) Let Z be a discrete random variable with E(Z) = 0. Does it necessarily follow that E(Z³) = 0? If yes, give a proof; if no, give a counterexample.