D.1.1. Notational Conventions in .NET Writer 39 barcode in .NET D.1.1. Notational Conventions

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D.1.1. Notational Conventions generate, create 3 of 9 none for .net projects PDF417 Throughout th .net framework Code 3 of 9 e entire book we refer only to discrete probability distributions. Speci cally, the underlying probability space consists of the set of all strings of a certain length , taken with uniform probability distribution.

That is, the sample space is the set of all -bit long strings, and each such string is assigned probability measure 2 . Traditionally,. APPENDIX D random variab Code 3/9 for .NET les are de ned as functions from the sample space to the reals. Abusing the traditional terminology, we also use the term random variable when referring to functions mapping the sample space into the set of binary strings.

We often do not specify the probability space, but rather talk directly about random variables. For example, we may say that X is a random variable assigned values in the set of all strings such that Pr[X = 00] = 1 4 and Pr[X = 111] = 3 . (Such a random variable may be de ned over the sample space 4 {0, 1}2 , so that X (11) = 00 and X (00) = X (01) = X (10) = 111.

) One important case of a random variable is the output of a randomized process (e.g., a probabilistic polynomialtime algorithm, as in Section 6.

1). All our probabilistic statements refer to random variables that are de ned beforehand. Typically, we may write Pr[ f (X ) = 1], where X is a random variable de ned beforehand (and f is a function).

An important convention is that all occurrences of the same symbol in a probabilistic statement refer to the same (unique) random variable. Hence, if B( , ) is a Boolean expression depending on two variables, and X is a random variable, then Pr[B(X, X )] denotes the probability that B(x, x) holds when x is chosen with probability Pr[X = x]. For example, for every random variable X , we have Pr[X = X ] = 1.

We stress that if we wish to discuss the probability that B(x, y) holds when x and y are chosen independently with identical probability distribution, then we will de ne two independent random variables each with the same probability distribution. Hence, if X and Y are two independent random variables, then Pr[B(X, Y )] denotes the probability that B(x, y) holds when the pair (x, y) is chosen with probability Pr[X = x] Pr[Y = y]. For example, for every two independent random variables, X and Y , we have Pr[X = Y ] = 1 only if both X and Y are trivial (i.

e., assign the entire probability mass to a single string). Throughout the entire book, Un denotes a random variable uniformly distributed over the set of all strings of length n.

Namely, Pr[Un = ] equals 2 n if {0, 1}n and equals 0 otherwise. We often refer to the distribution of Un as the uniform distribution (neglecting to qualify that it is uniform over {0, 1}n ). In addition, we occasionally use random variables (arbitrarily) distributed over {0, 1}n or {0, 1} (n) , for some function : N N.

Such random variables are typically denoted by X n , Yn , Z n , and so on. We stress that in some cases X n is distributed over {0, 1}n , whereas in other cases it is distributed over {0, 1} (n) , for some function (which is typically a polynomial). We often talk about probability ensembles, which are in nite sequences of random variables {X n }n N such that each X n ranges over strings of length bounded by a polynomial in n.

Statistical difference. The statistical distance (aka variation distance) between the random variables X and Y is de ned as 1 2 . Pr[X = v] Pr[Y = v]. = max{Pr[X .net framework bar code 39 S] Pr[Y S]}..

(D.1). We say that X Code 39 Extended for .NET is -close (resp., -far) to Y if the statistical distance between them is at most (resp.

, at least) ..
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