slides16 - Lecture Stat 302 Introduction to Probability -...

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Unformatted text preview: Lecture Stat 302 Introduction to Probability - Slides 16 AD March 2010 AD () March 2010 1 / 8 Change of Variables Let X be a r.v. of pdf f X ( x ) and consider the r.v. Y = g ( X ) . A legitimate question is to ask what is the pdf f Y ( y ) of Y . This has numerous applications: converting measurements, computing returns on investments etc. AD () March 2010 2 / 8 Example: Conversion Celsius to Farenheit Consider X the temperature at a given time instant in Celsius. It is assumed that X has a pdf f X ( x ) and associated distribution function F X ( x ) . Assume you want to convert this temperature in Fahrenheit, hence you introduce the r.v. Y = 9 5 X + 32 . What is the distribution F Y ( y ) and its associated density f Y ( y ) ? Consider the general case where Y = aX + b then for a > F Y ( y ) = Pr ( Y & y ) = Pr ( aX + b & y ) = Pr & X & y b a = F X & y b a ) f Y ( y ) = dF Y ( y ) dy = 1 a f X & y b a ....
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slides16 - Lecture Stat 302 Introduction to Probability -...

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