8Continuous_Random_Variables

8Continuous_Random_Variables - Chapter 8 Continuous Random...

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Chapter 8 Continuous Random Variables In our previous discussion, we introduced discrete random variables and dis- cussed their properties. Discrete random variables are quite useful in many contexts, yet they form only a small subset of the collection of random vari- ables pertinent to applied probability and engineering. In this chapter, we consider random variables that range over a continuum of possible values; that is, random variables that can take on an uncountable set of values. Continuous random variables are powerful mathematical abstractions that allow practitioners to pose and solve important engineering problems, which cannot be addressed using discrete models. While this extra Fexibility is useful and desirable, it comes at a certain cost. A continuous random variable cannot be characterized by a probability mass function. This di±culty emerges from the limitations of the third axiom of probability laws, which only applies to countable collections of disjoint events. Below, we provide a de²nition for continuous random variables. ³urther- more, we extend and apply the concepts and methods initially developed for discrete random variables to the class of continuous random variables. In par- ticular, we develop a continuous counterpart to the probability mass function. 8.1 Cumulative Distribution Functions We begin our exposition of continuous random variables by introducing a gen- eral concept which can be employed to bridge our understanding of discrete 89
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90 CHAPTER 8. CONTINUOUS RANDOM VARIABLES and continuous random variables. Recall that a random variable is a real- valued function acting on the outcomes of an experiment. In particular, given a sample space, random variable X is a function from Ω to R . The cumulative distribution function (CDF) of X is deFned point-wise as the probability of the event { X x } , F X ( x ) = Pr( { X x } ) = Pr( X x ) . In terms of the underlying sample space, F X ( x ) denotes the probability of the set of all outcomes in Ω for which the value of X is less than or equal to x , F X ( x ) = Pr ( X - 1 (( -∞ ,x ]) ) = Pr( { ω Ω | X ( ω ) x } ) . In essence, the CD± is a convenient way to specify the probability of all events of the form { X ( -∞ ] } . The CD± of random variable X exists for any well-behaved function X : Ω ±→ R . Moreover, since the realization of X is a real number, we have lim x ↓-∞ F X ( x ) = 0 lim x ↑∞ F X ( x )=1 . Suppose x 1 <x 2 , then we can write { X x 2 } as the union of the two disjoint sets { X x 1 } and { x 1 <X x 2 } . It follows that F X ( x 2 ) = Pr( X x 2 ) = Pr( X x 1 ) + Pr( x 1 x 2 ) Pr( X x 1 )= F X ( x 1 ) . (8.1) In other words, a CD± is always a non-decreasing function. ±inally, we note from (8.1) that the probability of X falling in the interval ( x 1 2 ] is Pr( x 1 x 2 F X ( x 2 ) - F X ( x 1 ) . (8.2) 8.1.1 Discrete Random Variables If X is a discrete random variable, then the CD± of X is given by F X ( x ± u X (Ω) ( -∞ ,x ] p X ( u ) ,
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8.1. CUMULATIVE DISTRIBUTION FUNCTIONS 91 and its PMF can be computed using the formula p X ( x ) = Pr( X x ) - Pr( X < x )= F X ( x ) - lim u x F X ( u ) .
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This note was uploaded on 03/30/2010 for the course ECEN 303 taught by Professor Chamberlain during the Fall '07 term at Texas A&M.

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8Continuous_Random_Variables - Chapter 8 Continuous Random...

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