Knowing the form of the cdfs pdfs or pmfs and

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Unformatted text preview: −1 ≤ u < 0 ↔ fY |X (v |u) = 0 ≤ u < 1 ↔ fY |X (v |u) = 2 1+u 1+u 2 0 else 2 1−u 1+u 2 0 else ≤v ≤1+u ≤v≤1 ↔ uniform on 1+u ,1 + u 2 ↔ uniform on 1+u ,1 , 2 As indicated, the distribution of Y given X = u for u fixed, is the uniform distribution over an interval depending on u, as long as −1 < u < 1. The mean of a uniform distribution over an interval is the midpoint of the interval, and thus: ∗ g (u) = E [Y |X = u] = 3(1+u) 4 3+u 4 −1 < u ≤ 0 0<u<1 undefined else. 4.10. MINIMUM MEAN SQUARE ERROR ESTIMATION 167 This estimator is shown in Figure 4.25(b). Note that this estimator could have been drawn by inspection. For each value of u in the interval −1 < u < 1, E [Y |X = u] is just the center of mass of the cross section of the support of fX,Y along the vertical line determined by u. That is true whenever (X, Y ) is uniformly distributed over some region. The mean square error given X = u 2 is the variance of a uniform distrib...
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