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In: Statistics and Probability

If the joint probability density function of the random variables X and Y is given by...

If the joint probability density function of the random variables X and Y is given by f(x, y) = (1/4)(x + 2y) for 0 < x < 2, 0 < y < 1, 0 elsewhere

(a) Find the conditional density of Y given X = x, and use it to evaluate P (X + Y/2 ≥ 1 | X = 1/2)

(b) Find the conditional mean and the conditional variance of Y given X = 1/2

(c) Find the variance of W = X − 2Y + 3

(d) Find the covariance of X and Y, and determine if X and Y are independent

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