HW08_2011 - N (0 ; 2 ) random variables. (1) Find the joint...

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PROF HONG FALL 2011 ECONOMICS 6190 PROBLEM SET # 8 1. [# 4.10, p.193] The random pair ( X;Y ) has the distribution X 1 2 3 2 1 12 1 6 1 12 Y 3 1 6 0 1 6 4 0 1 3 0 (1) Show that X and Y are dependent. (2) Give a probability table for random variables U and V that have the same marginals as X and Y but are independent. 2. [# 4.17, p.194] Let X Y to be the integer part of X + 1 , that is Y = i + 1 if and only if i X < i + 1 ;i = 0 ; 1 ; 2 ;::: (1) Find the distribution of Y . What well-known distribution does Y have? (2) Find the conditional distribution of X ± 4 given Y ² 5 : 3. [#4.19, p.194] (1) Let X 1 and X 2 be independent N (0,1) random variables. Find the pdf of ( X 1 ± X 2 ) 2 = 2 : (2) If X i ;i = 1 ; 2 ; are independent Gamma( i ; 1 marginal distributions of X 1 = ( X 1 + X 2 ) and X 2 = ( X 1 + X 2 ) : 4. Suppose X 1 ³ Gamma ( 1 ; 1) ; X 2 ³ Gamma ( 2 ; 1) ; and X 1 and X 2 are independent. Show that X 1 + X 2 and X 1 = ( X 1 + X 2 ) X 1 + X 2 and X 1 = ( X 1 + X 2 ) ; respectively. 5. [#4.20, p.194] X 1 and X 2 are independent
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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 Beta ( &amp;; ) ; and Y Beta ( &amp; + ; ) be independent random variables, &amp;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 &amp; N ( &amp;; 2 ) , and let Y &amp; N ( ; 2 ) : Suppose X and Y are independent. De&amp;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 &amp; N (0 ; 1) ;X 2 &amp; N (0 ; 1) and X 1 and X 2 are independent. Find the distribution of X 1 =X 2 . 2...
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HW08_2011 - N (0 ; 2 ) random variables. (1) Find the joint...

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