12 proposition 315 a function f is the cdf of some

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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 discrete-type and continuous-type 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. CONTINUOUS-TY...
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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.

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