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Atlanta City Hall Annex Atlanta City Hall Tower MonFri 815 am to 500 pmIf X and Y are independent, then E(es(XY )) = E(esXesY) = E(esX)E(esY), and we conclude that the mgf of an independent sum is the product of the individual mgf's Sometimes to stress the particular rv X, we write M X(s) Then the above independence property can be concisely expressed as M XY (s) = M X(s)M Y (s), when X and Y are independent In general, we recommend citing the original references describing the theoretical methods used when reporting results obtained from Gaussian calculations, as well as giving the citation for the program itself These references are given in the discussions of the relevant keywords The only exceptions occur with long established methods such as
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Electromagnetic energy flow the rate at which energy flows through a surface S is GG given by P = ∫ S da ⋅ S Useful Math Cartesian Gradient ∇t = ∂ t xˆ ∂ t yˆ ∂ t zˆ ∂ x ∂ y ∂ z Divergence ∇ ⋅ v z G = ∂ vx ∂ vy ∂ v ∂ x ∂ y ∂ z G ∂ vy ∂ v Curl ∇ × v = (x x ∂ vz −X,y p(x,y)logp(x,y) (4) The joint entropy measures how much uncertainty there is in the two random variables X and Y taken together Definition The conditional entropy of X given Y is H(XY) = − X x,y p(x,y)logp(xy) = −E log(p(xy)) (5) The conditional entropy is a measure of how much uncertainty remains about the random variable0 0 1 2 3 / 4 * $ * 5 3 & 0 1 23 4 5 6 7 8 9 5 ;



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