2. Let X and Y be continuous random variables having the joint density function: [c(x² + y²) 0 f(x, y) = 0≤x≤1, 0≤y≤1 otherwise Determine a) The constant c b) P(X< 1/2, Y> 1/2) c) P(1/4 < X < 3/4) d) P(Y<1/2) e) The conditional distribution of X given Y=y f) Whether or not X and Y are independent.
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a. The constant c is 3/2
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- Let X and Y be a pair of continuous random variables with a joint density fx,y(x,y). Assume that fx,y(x,y) = cxy for x greater than or equal to 0, y greater than or equal to 0, and x + y less than or equal to 1. Here c is a constant. Assume that fx,y(x,y) is 0 elsewhere. What is the constant c equal to? With the value of c, what is E[XY]?Suppose that X, Y , and Z are random variables with a joint density f(x, y, z) = ( 6/((1+x+y+z)^4)) , when x, y, z > 0, and 0, otherwise. Determine the distribution of X + Y + Z.Suppose that the random variables X and Y have a joint density function given by: f(x,y) = {c(2x+y) for 2≤x≤6 and 0≤y≤5, 0 otherwise P(3 < X < 5, Y >1), P(X < 3), P(X +Y > 5), Find the joint distribution function (cdf),
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