In Section 3.7, the joint density of the minimum and maximum of n independent uniform random variables was found. In the case n = 2, this amounts to X and Y , the minimum and maximum, respectively, of two independent random variables uniform on [0, 1], having the joint density
f ( x , y ) = 2, 0 ≤ x ≤ y
a. Find the covariance and the correlation of X and Y . Does the sign of the correlation make sense intuitively?
b. Find E(X|Y = y) and E(Y |X = x).Dothese resultsmake sense intuitively?
c. Find the probability density functions of the random variables E(X|Y ) and E(Y |X).
d. What is the linear predictor of Y in terms of X (denoted by ˆY = a + bX) that has minimal mean squared error? What is the mean square prediction error?
e. What is the predictor of Y in terms of X [ˆY = h(X)] that has minimal mean squared error? What is the mean square prediction error?
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