Unformatted text preview: a given application often one way to express the distribution is more
useful than another. 74 3.2 CHAPTER 3. CONTINUOUSTYPE RANDOM VARIABLES Continuoustype random variables Deﬁnition 3.2.1 A random variable X is a continuoustype random variable if there is a function
fX , called the probability density function (pdf ) of X , such that
c fX (u)du, FX (c) =
−∞ for all c ∈ R. The support of a pdf fX is the set of u such that fX (u) > 0.
It follows by the fundamental theorem of calculus that if X is a continuoustype random variable
then the pdf is the derivative of the CDF: fX = FX . In particular, if X is a continuoustype random
variable, FX is diﬀerentiable, and therefore FX is a continuous function. That is, there are no jumps
in FX , so for any constant v , P {X = v } = 0.
It may seem strange at ﬁrst that P {X = v } = 0 for all numbers v , because if we add these
probabilities up over all v , we get zero. It would seem like X can’t take on any real value. But remember that there are uncountably many real numbers, and th...
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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 Tech.
 Spring '08
 Zahrn
 The Land

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