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**Unformatted text preview: **} = F (x). It now follows from (ii) and the fact that P is a probability on (R, B ) that
lim F (x) = P {(−∞, −∞)} = P {∅} = 0 x→−∞ and
lim F (x) = P {(−∞, ∞)} = P {R} = 1. x→∞ This establishes (i). To show that F is increasing, observe that if x ≤ y , then (−∞, x] ⊆
(−∞, y ]. Since P is a probability, this implies that F (x) = P {(−∞, x]} ≤ P {(−∞, y ]} =
F (y ). This establishes (iii) and taken together the proof is complete.
11–1 A ﬁrst look at random variables
Consider a chance experiment. We have deﬁned a probability space (Ω, F , P) consisting of a
sample space Ω of outcomes, a σ -algebra F of events, and an assignment P of probabilities
to events as a model for the experiment. It is often the case that one is not interested in a
particular outcome per se, but rather in a function of the outcome. This is readily apparent
if we consider a bet on a game of chance at a casino. For instance, suppose that a gambler
pays $3 to roll a fair die and then wins $j where j is the side that appears, j = 1, . . . , 6.
Hence, the gambler’s net income is either −$2, −$1, $0, $1, $2, or $3 depending on whether
a 1, 2, 3, 4, 5, or 6 appears. If we let Ω = {1, 2, 3, 4, 5, 6} denote the sample space for this
experiment, and we let X denote the gambler’s net income, then it is clear that X is the
real-valued function on Ω given by
X (1) = −2, X (2) = −1, X (3) = 0, X (4) = 1, X (5) = 2,...

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