NotesOct25

# NotesOct25 - Example joint pdf Let(X1 X2 be continuous with...

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Example Let ( X 1 , X 2 ) be continuous with a joint pdf f X 1 ,X 2 ( x 1 , x 2 ) = ( 1 if 0 < x 1 , x 2 < 1 0 otherwise. Let Y 1 = X 1 + X 2 , Y 2 = X 1 - X 2 . Find the joint pdf of ( Y 1 , Y 2 ).

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Standard transformations The sum of independent random variables Let X 1 and X 2 be independent continuous ran- dom variables with densities f X 1 and f X 2 ac- cordingly. Let Z = X 1 + X 2 . The density of Z is f Z ( z ) = Z -∞ f X 1 ( z - v ) f X 2 ( v ) dv, the convolution of the densities

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Unformatted text preview: f X 1 and f X 2 . Example : Let X and Y be iid exponential ran-dom variables with parameter λ . Find the pdf of the sum Z = X + Y . • The diﬀerence of independent random vari-ables Z = X 1-X 2 : The density of Z : f Z ( z ) = Z ∞-∞ f X 1 ( z + v ) f X 2 ( v ) dv....
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• '10
• SAMORODNITSKY
• Probability distribution, Probability theory, probability density function, X1, independent random variables

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