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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 ] œ 1Ð\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] Ñ œ \ œ ] œ #C 0Ð1 ÐCÑÑ œ #/ " " #C # .B .C # 0 ÐCÑ œ #C † #/ M ÐC Ñ ] #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 1 Ð] Ñ œ \ œ ] œ C 0Ð1 ÐCÑÑ œ M ÐC Ñ œ M ÐCÑÞ " #Î$ " "Î$ "Î$ .B .C $ 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 ^ 0 0Ð Ñ œ Ð# Ñ / M ÐDÑÞ 1 "Î# D Î# Ò_ß_Ó # Find the pdf for ] œ 1Ð] Ñ œ ^ 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
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