# m it remains to note that for every distortion

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Unformatted text preview: in assuming that minj d(i, j ) = 0 for all i ∈ {1, 2, . . . , m}, it remains to note that for every distortion measure d(·, ·), there exists a corresponding distortion measure d (·, ·) with minj d (i, j ) = 0 for every i ∈ {1, 2, . . . , m} (by choosing wi = minj d(i, j )). And as we have shown, the rate distortion function R(·) then follows directly from R (·) by R(D ) = R (D − w ). ¯ c Amos Lapidoth, 2012 1 Erasure Distortion Problem 2 Note ﬁrst that for rate R = 0 the distortion D = 1 is achievable through the choice PX |X (“?”|x) = 1. ˆ Furthermore, d(1, 0) = ∞ and d(0, 1) = ∞ imply that the rate distortion function is certainly only ˆ ˆ achieved by conditional laws satisfying Pr X = 0 X = 1 = 0 and Pr X = 1 X = 0 = 0, since otherwise the expected distortion would be inﬁnite. It follows that ˆ E d(X, X ) = ˆ Pr X = x X = x d(x, x) ˆ ˆ Pr[X = x] x 1 = 2 x ˆ ˆ ˆ ˆ Pr X = 0 X = 0 · 0 + Pr X = “?” X = 0 · 1 + Pr X = 1 X = 1 · 0 ˆ + Pr X = “?” X = 1 · 1 = 1 2 ˆ ˆ Pr X = “?” X = 0 + Pr X = “?...
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