Lecture713-04 - Functions of Random Variables Lecture 4...

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Functions of Random Variables Lecture 4 Spring 2002
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Function of a Random Variable Let U be an random variable and V = g ( U ). Then V is also a rv since, for any outcome e , V ( e ) = g ( U ( e )). There are many applications in which we know F U ( u ) and we wish to calculate F V ( v ) and f V ( v ). The distribution function must satisfy F V ( v ) = P [ V v ] = P [ g ( U ) v ] To calculate this probability from F U ( u ) we need to find all of the intervals on the u axis such that g ( u ) v . Lecture 4 1
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Function of a Random Variable v v 1 if u a v v 2 if u b or c u d v v 3 if u e For any number s , values of u such that g ( u ) s fall in a set of intervals I s . Lecture 4 2
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Example: V = aU + b For any s , t = s b a defines the interval I s = { u : u t } = u : u s b a For any probability distribution function F U ( u ) we then find F V ( v ) = P U v b a = F U v b a Lecture 4 3
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Example: V = aU + b Suppose U has a uniform distribution on the interval 1 u 1.
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  • Fall '08
  • TomMazzuchi
  • Probability distribution, Probability theory, probability density function, Cumulative distribution function

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