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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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 Spring '08
 Zahrn
 The Land

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