5 ways to describe distributions of random variables

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Unformatted text preview: of an arbitrary random variable X determines the probabilities of any events of the form {X ∈ A}. Of course, the CDF of a random variable X determines P {X ≤ c} for any real number c–by deﬁnition it is just FX (c). Similarly, P {X ∈ (a, b]} = FX (b) − FX (a), whenever a < b. The next proposition explains how FX also determines probabilities of the form P {X < c} and P {X = c}. Proposition 3.1.2 Let X be a random variable and let c be any real number. Then P {X < c} = FX (c−) and P {X = c} = FX (c), where FX is the CDF of X. Proof. Fix c and let c1 , c2 , . . . be a sequence with c1 < c2 < . . . such that limj →∞ cj = c. Let G1 be the event G1 = {X ≤ c1 } and for j ≥ 2 let Gj = {cj −1 < X ≤ cj }. Then for any n ≥ 1, {X ≤ cn } = G1 ∪ G2 ∪ · · · ∪ Gn . Also, {X < c} = G1 ∪ G2 ∪ · · · and the events G1 , G2 , . . . are mutually exclusive. Therefore, by Axiom P.2, P {X < c} = P (G1 ) + P (G2 ) + · · · . 3.1. CUMULATIVE DISTRIBUTION FUNCTIONS 71 The sum of a series is, by deﬁ...
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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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