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Unformatted text preview: is violated. (a) 1 (d) 1 (b) (e) (c) 1 1 (f) 1 1 Figure 3.4: Six candidate CDFs. Solution: The functions shown in plots (a), (c), and (f) are valid CDFs and the other three are
not. The function in (b) is not nondecreasing, and it does not converge to zero at −∞. The function
in (d) does not converge to zero at −∞. The function in (e) is not right continuous.
The vast majority of random variables described in applications are one of two types, to be
described next. A random variable X is a discretetype random variable if there is a ﬁnite or 3.1. CUMULATIVE DISTRIBUTION FUNCTIONS 73 countably inﬁnite set of values {ui : i ∈ I } such that P {X ∈ {ui : i ∈ I }} = 1. The probability
mass function (pmf) of a discretetype random variable X , denoted pX (u), is deﬁned by pX (u) =
P {X = u}. Typically the pmf of a discrete random variable is much more useful than the CDF.
However, the pmf and CDF of a discretetype random variable are related by pX (u) = FX (u)
and conversely,
pX (u), FX (c) = (3.1) u:u≤c where the sum in (3.1) is taken only over u such that pX (u) = 0. If X is a discretetype random
variable with only ﬁnitely many m...
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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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