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Suppose X is normally distributed with mean 1 and standard deviation 0.25, and Y...

Suppose X is normally distributed with mean 1 and standard deviation 0.25, and Y is also normal with mean 1.5 and standard deviation 0.4. Suppose that X and Y have correlation coefficient 0.6. Find the following probabilities: (a) P(X 2 1.3) (b) P(X+y-2.5) (c) P(X +Y 2 3) (d) P(Y - X so) (e) P(Y <X)

Answer 1

a) Z = x-mu i/sigma is standard normally distributed r.v with mean 0, variance 1. p(x greaterthanorequalto 1.3) = p(z greaterthanorequalto 1.3-mu i/sigma) = p (z greaterthanorequalto 1.3 - 1/0.25) = p (z greaterthanorequalto 1.2) = 0.5 - 0.3849 = 0.1151. b) correlation coefficient = COV (x, y)/sigma_x sigma_y cov(x, y) = 0.6 times 0.25 times 0.4 = 0.06. E[x + y] = E[X] + E[Y] = 1 + 1.5 = 2.5 var[X + Y] = var[X] + var[Y] + 2 cov (X, Y) = (0.25)^2 + (0.4)^2 + 2 times 0.06 = 0.3425 p(x + y greaterthanorequalto 2.5) = p (z greaterthanorequalto 2.5 - 2.5/squarerootof 0.3425) = p (z greaterthanorequalto 0) = 0.5 c) p (x + y greaterthanorequalto 3) = p (z greaterthanorequalto 0.5/ squarerootof 0.3425) = p(z greaterthanorequalto 0.8544) = 0.5 - 0.3035 = 0.1965 E[Y - X] = E[Y] - E[X] = 1.5 - 1 = 0.5 var[Y - X] = var [Y] + var [X] - 2 COV(X, Y) = (0.25)^2 + (0.4)^2 - 2 times 0.06 = 0.1025 p (y - x lessthanoreualto 0)