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Unformatted text preview: a given application often one way to express the distribution is more useful than another. 74 3.2 CHAPTER 3. CONTINUOUS-TYPE RANDOM VARIABLES Continuous-type random variables Definition 3.2.1 A random variable X is a continuous-type 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 continuous-type random variable then the pdf is the derivative of the CDF: fX = FX . In particular, if X is a continuous-type 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.

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