Let X be a random variable with density In(1+x) 1+x 1[0,1] (x) 2 x(x) = (ln(2))...

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Let X be a random variable with density In(1+x) 1+x 1[0,1] (x) 2 x(x) = (ln(2)) Let y be a random variable such that the conditional density of Y given X = x is given by fy|x=x(y) = = 1 In(1+x) 1[0,x] (y) 1+y . 1. Give the joint density of the pair (X, Y). 2. Are the variables X and Y independent? 3. What is the law of Y? What is the conditional law of X given Y? 4. Determine the conditional expectation E[Y|X]. 5. Determine the conditional expectation E[X|Y]. Let X be a random variable with density In(1+x) 1+x 1[0,1] (x) 2 x(x) = (ln(2)) Let y be a random variable such that the conditional density of Y given X = x is given by fy|x=x(y) = = 1 In(1+x) 1[0,x] (y) 1+y . 1. Give the joint density of the pair (X, Y). 2. Are the variables X and Y independent? 3. What is the law of Y? What is the conditional law of X given Y? 4. Determine the conditional expectation E[Y|X]. 5. Determine the conditional expectation E[X|Y].