Isye 2027

# Let be the set of all such s the set of events can be

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Unformatted text preview: + x + x + ··· = 1−x 2 ∞ xk = or k=0 1 . 1−x (1.4) The formula (1.3) for partial sums and the formula (1.4) for inﬁnite sums of a geometric series are used frequently in these notes. 3 “ω ” is a lower case omega. 1.2. AXIOMS OF PROBABILITY 11 Example 1.2.4 (Repeated binary trials) Suppose we would like to represent an inﬁnite sequence of binary observations, where each observation is a zero or one with equal probability. For example, the experiment could consist of repeatedly ﬂipping a fair coin, and recording a one each time it shows heads and a zero each time it shows tails. Then an outcome ω would be an inﬁnite sequence, ω = (ω1 , ω2 , · · · ), such that for each i ≥ 1, ωi ∈ {0, 1}. Let Ω be the set of all such ω ’s. The set of events can be taken to be large enough so that any set that can be deﬁned in terms of only ﬁnitely many of the observations is an event. In particular, for any binary sequence (b1 , · · · , bn ) of some ﬁnite length n, the set {ω ∈ Ω : ωi = bi for 1 ≤ i ≤ n} should be in...
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## This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Tech.

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