By Voloshynovskiy, Herrigel, Baumgaertner, Pun

Show description

Read Online or Download A Stochastic Approach to Content Adaptive Digital Image Watermarking PDF

Best probability books

Introduction to Probability Models (10th Edition)

Ross's vintage bestseller, advent to chance types, has been used broadly by means of professors because the fundamental textual content for a primary undergraduate direction in utilized likelihood. It offers an advent to straightforward chance thought and stochastic techniques, and indicates how chance idea will be utilized to the learn of phenomena in fields similar to engineering, machine technological know-how, administration technological know-how, the actual and social sciences, and operations learn.

Real analysis and probability

This vintage textbook, now reissued, bargains a transparent exposition of recent likelihood idea and of the interaction among the homes of metric areas and likelihood measures. the recent variation has been made much more self-contained than sooner than; it now features a beginning of the genuine quantity method and the Stone-Weierstrass theorem on uniform approximation in algebras of services.

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

Sample text

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, .

Download PDF sample

Rated 4.70 of 5 – based on 42 votes