By Voloshynovskiy, Herrigel, Baumgaertner, Pun

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Extra resources for A Stochastic Approach to Content Adaptive Digital Image Watermarking

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K! λk λk −λ npn n (1 − ) → e . k! n k! In the second step we used that (1 − pn )k → 1; the last convergence follows from the well-known approximation formula for the exponential function. 32) we will see that the deviation from the limit is of order 1/n. 17) sums up to 1, and thus defines a discrete density on Z+ . The corresponding probability measure is one of the fundamental distributions in stochastics. Definition. For λ > 0, the probability measure Pλ on (Z+ , P(Z+ )) defined by Pλ ({k}) = e−λ λk /k!

Hence F is the distribution function of X. ✸ Since every probability measure P on (R, B) is uniquely determined by its distribution function, we can rephrase the proposition as follows: Every P on (R, B ) is the distribution of a random variable on the probability space (]0, 1[, B]0,1[ , U]0,1[ ). This fact will repeatedly be useful. The connection between distribution functions and probability densities is made by the notion of a distribution density. 31) Remark and Definition. Existence of a distribution density.

N} with the pair (n − k, k) ∈ . Setting p = (1) ∈ [0, 1], we find that the multinomial distribution Mn, is reduced to the binomial distribution Bn, p on {0, . . , n} with density Bn, p ({k}) = n k p (1 − p)n−k , k ∈ {0, . . , n}. 1. As a summary of this section we thus obtain the following. 9) Theorem. Multinomial distribution of the sampling histogram. Suppose E is a finite set with |E| ≥ 2, a discrete density on E and P = ⊗n the associated n-fold product measure on = E n . 7) then has the multinomial distribution P ◦ S −1 = Mn, .