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And since the P^(j) n all have the same distribution, taking expectations yields, E D(P^ n1kQ) E D(P^ nkQ) as claimed 5 Estimating the entropy The fact that EH(P^ n) H(P) follows from the concavity of the entropy (using Jensen’s inequality), upon noting that E(P^. 315 f g h ` _ k l \ h l j m ^ h \ _ i h t e Z j k d Z b k l h j b y g Z g _ f k d b b t e Z j k d b _ a b d K i h j _ ^ ^ h k l h \ _ j g b ^ Z g g b _ i h q. !"#$%&’( )*,/0123!" 0 1 2 3 4 5 6 7 8 9 8;.

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Variable In other words, if Y(= lnX) is normally distributed, eY(= X) is lognormally distributed µ Y = µ lnX, σ 2 Y = σ 2 X, PY ≤y PlnX≤lnx PX≤x = F Y(y) F lnX(lnx) F X(x) = Φ y−µ Y σ Y Φ lnx−µ lnX σ lnX The mean and standard deviation of a lognormal variable Xare related to the mean and standard deviation of lnX. Imize) f(x,y,z) subject to the constraints g(x,y,z) = 0 and h(x,y,z) = 0” We use the technique of Lagrange multipliers To do so, we define the auxiliary function L(x,y,z,λ,µ) = f(x,y,z)λg(x,y,z)µh(x,y,z) It is a function of five variables — the original variables x, y and z, and two auxiliary variables λ and µ. ñ Ç } µ l v } Á Z d } o } u u © Z } À v P o Z v P } v } Z.

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