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Unformatted text preview: random variable is through the use
of a cumulative distribution function, as described in the next section. Cumulative distribution
functions form a natural bridge between discretetype and continuoustype random variables. 3.1 Cumulative distribution functions Let a probability space (Ω, F , P ) be given. Recall that in Chapter 2 we deﬁned a random variable
to be a function X from Ω to the real line R. To be on mathematically ﬁrm ground, random
variables are also required to have the property that sets of the form {ω : X (ω ) ≤ c} should be
events–meaning that they should be in F . Since a probability measure P assigns a probability to
every event, every random variable X has a cumulative distribution function (CDF), denoted by
FX . It is the function, with domain the real line R, deﬁned by
FX (c) = P {ω : X (ω ) ≤ c}
= P {X ≤ c} (for short).
Example 3.1.1 Let X denote the number showing for a roll of a fair die, so the pmf of X is
1
pX (i) = 6 for integers i with 1 ≤ i ≤ 6. The CDF FX is shown in Figure 3.1. 69 70 CHAPTER 3. CONTINUOUSTY...
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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 Institute of Technology.
 Spring '08
 Zahrn
 The Land

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