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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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 Spring '08
 Zahrn
 The Land

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