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Here the coin flip could be modeled in a directly given fashion by just describing the sample space and the probability measure, or it can be modeled in an indirect fashion 22 CHAPTER 2. PROBABILITY as a function (signal processing, random variable) on another experiment. This suggests, for example, that to study coin flips empirically we could either actually flip a fair coin, or we could spin a fair wheel and quantize the output. Although the second method seems more complicated, it is in fact extremely common since most random number generators (or pseudorandom number generators) strive to produce random numbers with a uniform distribution on [0, 1) and all other probability measures are produced by further signal processing.

Then many countably infinite sets of binary sequences (say the set of all periodic sequences) are not events since they cannot be expressed as finite sequences of set-theoretic operations on the singleton sets. Obviously, the sigma-field formed by including countable set-theoretic operations does not have this defect. This is why sigma-fields must be used rather than fields. 3. 19) can be related to a condition on limits by defining the notion of a limit of a sequence of sets. This notion will prove useful when interpreting the axioms of probability.

Let {Fki ; i = 0, 1, . . , K − 1} be a collection of members of F. Then a set of the form {{xt ; t ∈ I} : xki ∈ Fki ; i = 0, 1, . . , K − 1} is an example of a finite-dimensional set. Note that it collects all sequences or waveforms such that a finite number of coordinates are constrained to lie in one-dimensional events. 3(d). Observe that when the one-dimensional sets constraining the coordinates are intervals, then the two-dimensional sets are rectangles. Analogous to the two-dimensional example, finite-dimensional events having separate constraints on each coordinate are called rectangles.

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An Introduction to Statistical Signal Processing last edition by Gray Robert

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