By Robert Gray

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**Extra info for An Introduction to Statistical Signal Processing new edition**

**Example text**

Let A be a sample space and let Ai be replicas or copies of A. We will consider both one-sided and two-sided infinite products to model sequences with and without a finite origin, respectively. Define the two-sided space Ai = { all sequences {ai ; i = . . , −1, 0, 1, . }; ai ∈ Ai } , i∈Z and the one-sided space Ai = { all sequences {ai ; i = 0, 1, . }; ai ∈ Ai } . i∈Z+ ∞ ∞ These two spaces are also denoted by i=−∞ Ai or ×∞ i=−∞ Ai and i=0 Ai or ×∞ A , respectively. i i=0 The two spaces under discussion are often called sequence spaces.

This example is different from the spinning wheel in that the sample space is discrete instead of continuous and that the probabilities of events are defined by sums instead of integrals, as one should expect when doing discrete math. 9) hold in this case as well (since sums behave like integrals), which in turn implies that the simple properties (a)–(d) also hold. A Single Coin Flip as Signal Processing The coin flip example can also be derived in a very different way that provides our first example of signal processing.

PROBABILITY SPACES 25 in the former and element inclusion in the latter space are clear. Consider especially the difference between an element of Ω and a subset of Ω that consists of a single point. The latter might or might not be an element of F, the former is never an element of F. Although the difference might seem to be merely semantics, the difference is important and should be thoroughly understood. A measurable space (Ω, F) is a pair consisting of a sample space Ω and an event space or sigma-field F of subsets of Ω.

### An Introduction to Statistical Signal Processing new edition by Robert Gray

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