Stat 351 Fall 2007 Assignment #7 This assignment is due at the ...
Stat 351 Fall 2007 Assignment #7 This assignment is due at the ...
Stat 351 Fall 2007 Assignment #7 This assignment is due at the ...
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<strong>St<strong>at</strong></strong> <strong>351</strong> <strong>Fall</strong> <strong>2007</strong><strong>Assignment</strong> <strong>#7</strong><strong>Th<strong>is</strong></strong> <strong>assignment</strong> <strong>is</strong> <strong>due</strong> <strong>at</strong> <strong>the</strong> beginning of class on Friday, November 9, <strong>2007</strong>. You must submitsolutions to all problems. As indic<strong>at</strong>ed on <strong>the</strong> course outline, solutions will be graded for bothcontent and clarity of exposition. The solutions th<strong>at</strong> you submit must be ne<strong>at</strong> and orderly. Donot crowd your work or write in multiple columns. Your <strong>assignment</strong> must be stapled and problemnumbers clearly labelled.1. Suppose th<strong>at</strong> X 1 and X 2 are independent N(0, 1) random variables. Set Y 1 = X 1 − 3X 2 + 2and Y 2 = 2X 1 − X 2 − 1.(a) Determine <strong>the</strong> d<strong>is</strong>tributions of Y 1 and Y 2 .(b) Determine <strong>the</strong> d<strong>is</strong>tribution of Y = (Y 1 , Y 2 ) ′ .2. Let X have a three-dimensional normal d<strong>is</strong>tribution with mean vector µ and covariancem<strong>at</strong>rix Λ given by⎛ ⎞⎛ ⎞32 1 3µ = ⎝ 4 ⎠ and Λ = ⎝1 4 −2⎠ ,−33 −2 8respectively. If Y 1 = X 1 + X 3 and Y 2 = 2X 2 , determine <strong>the</strong> d<strong>is</strong>tribution of Y = (Y 1 , Y 2 ) ′ .3. Suppose th<strong>at</strong> Y 1 , Y 2 , and Y 3 are independent N(0, 1) random variables. SetX 1 = Y 1 − Y 3 ,X 2 = 2Y 1 + Y 2 − 2Y 3 ,X 3 = −2Y 1 + 3Y 3 .Determine <strong>the</strong> d<strong>is</strong>tribution of X = (X 1 , X 2 , X 3 ) ′ .4. Suppose th<strong>at</strong> X 1 , X 2 , and X 3 are independent N(0, 1) random variables. SetY 1 = X 2 + X 3 ,Y 2 = X 1 + X 3 ,Y 3 = X 1 + X 2 .Determine <strong>the</strong> d<strong>is</strong>tribution of Y = (Y 1 , Y 2 , Y 3 ) ′ .(continued)
5. Suppose th<strong>at</strong> X ∈ U(0, 1), and th<strong>at</strong> <strong>the</strong> d<strong>is</strong>tribution of Y conditioned on X = x <strong>is</strong> N(x, x 2 );th<strong>at</strong> <strong>is</strong>, Y |X = x ∈ N(x, x 2 ).(a) Find E(Y ), var(Y ), and cov(X, Y ). Hint: Conditional expect<strong>at</strong>ions will simplify calcul<strong>at</strong>ions.(b) Prove th<strong>at</strong> Y/X and X are independent.(c) Use your results of (a) and (b) to show th<strong>at</strong>( ) YEXNote th<strong>at</strong> in general E(Y/X) ≠ E(Y )/E(X).= E(Y )E(X) .6. Suppose th<strong>at</strong>X = (X, Y ) ′ ∈ NShow th<strong>at</strong> <strong>the</strong> correl<strong>at</strong>ion of X 2 and Y 2 <strong>is</strong> ρ 2 .(( 00),( )) 1 ρ.ρ 1Some Hints: (i) If Z ∈ N(0, 1), use <strong>the</strong> moment gener<strong>at</strong>ing function to calcul<strong>at</strong>e E(Z 4 ).(ii) In order to calcul<strong>at</strong>e <strong>the</strong> higher moment involving X and Y , using conditional expect<strong>at</strong>ionswill gre<strong>at</strong>ly simplify <strong>the</strong> calcul<strong>at</strong>ion. Determine <strong>the</strong> d<strong>is</strong>tribution of Y |X. (Use Equ<strong>at</strong>ion (6.2) onpage 130.) Then use Theorem III.2.2 on page 37.