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slides13 - Lecture Stat 302 Introduction to Probability...

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Lecture Stat 302 Introduction to Probability - Slides 13 AD March 2010 AD () March 2010 1 / 12
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Continuous Random Variables For the time being, we have only considered discrete random variables (r.v.) - set of possible values is °nite or countable - such as the number of arrivals in a given time instants, the number of successful trials, the number of items with a given characteristic. In many practical applications, we have to deal with random variables whose set of possible values is uncountable. Example: Waiting for a bus . The time (in minutes) which elapses between arriving at a bus stop and a bus arriving can be modelled as a r.v. X taking values in [ 0 , ) . Example: Share price. The values of one share of a speci°c stock at some given future time can be modelled as a r.v. X taking values in [ 0 , ) . Example: Weight . The weight of a randomly chosen individual can be modelled as a r.v. X taking values in [ 0 , ) . Example: Temperature. The temperature in Celsius at a given time can be modelled as a r.v. X taking values in [ ° 273 . 15 , ) . AD () March 2010 2 / 12
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Continuous Random Variable °Formal±de²nition : We say that X is a (real-valued) continuous r.v. if there exists a nonnegative function f : R ! [ 0 , ) such that for any set A of real numbers P ( X 2 A ) = Z A f ( x ) dx . f ( x ) is called the probability density function (pdf) of the r.v. X and the associated (cumulative) distribution function is F ( x ) = Pr ( X ± x ) = Z
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