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Unformatted text preview: N (0 ;± 2 ) random variables. (1) Find the joint distribution of Y 1 and Y 2 , where Y 1 = X 2 1 + X 2 2 and Y 2 = X 1 = p Y 1 : (2) Show that Y 1 and Y 2 are independent. 6. [#4.23, p.195] For X ³ Beta ( &;² ) ; and Y ³ Beta ( & + ²;³ ) be independent random variables, &nd the distribution of XY by making the transformation given in (1) and (2) and integrating out V (1) U = XY;V = Y (2) U = XY;V = X=Y 1 7. [#4.27, p.195] Let X & N ( &;± 2 ) , and let Y & N ( ²;± 2 ) : Suppose X and Y are independent. De&ne U = X + Y and V = X ± Y . Show that U and V are independent normal random variables. Find the distribution of each of them. 8. Suppose X 1 & N (0 ; 1) ;X 2 & N (0 ; 1) and X 1 and X 2 are independent. Find the distribution of X 1 =X 2 . 2...
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 Fall '07
 HONG
 Economics, Econometrics, Probability theory, probability density function, x1 x2

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