print5 - Transformations of Continuous Random Variables I....

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Transformations of Continuous Random Variables I. Univariate case. If is a continuous random variable with pdf and is a continuous, invertible \0 1 function, then the pdf for is given by ]œ1 Ð\ Ñ 0 ÐCÑ œ l l 0Ð1 ÐCÑÑÞ ] .B .C " Example 1 . Let have pdf given by Find the pdf for \ 0 0ÐBÑ œ #/ M ÐBÑÞ ] œ #B Ò!ß_Ñ È Since , and , we obtain 1 Ð]Ñœ\œ] œ#C 0Ð1 ÐCÑÑœ#/ " " #C #. B # C Ñœ# C †# / M Ð ] #C # Ò!ß_Ñ # œ %C/ M ÐCÑÞ #C Ò!ß_Ñ # Example 2. Let have pdf given by 1 Find the pdf for \ 0 0ÐBÑ œ M ÐBÑÞ ] œ \ Þ Ò!ß $ 1] Since , we have and 1 œ C 0Ð1 ÐCÑÑœ M ÐC ÑœM ÐCÑ Þ " #Î$ " "Î$ "Î$ .B $ Ò!ß Ò!ß"Ó 1 1] Thus 0 ÐCÑ œ C M ÐCÑÞ ] $ #Î$ Ò!ß"Ó 1 Example 3. Let have pdf given by z ^00 Ð Ñ œ Ð # Ñ / M Ð D Ñ Þ 1 "Î# D Î# Ò_ß_Ó # Find the pdf for Ð]Ñœ^Þ # First note that is not invertible over the support of . We need to partition the support into 1^ the set of negative reals and the set nonnegative reals and write z 0Ð Ñ œ Ð# Ñ / M
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This note was uploaded on 01/02/2010 for the course APSC 116 taught by Professor Tommazzuchi during the Fall '08 term at GWU.

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print5 - Transformations of Continuous Random Variables I....

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