hw6 - Stats 116 HW6 1 Chapter 6 59 P Y 1 = i 1 Y k 1 = i k...

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Unformatted text preview: Stats 116: HW6 August 12, 2007 1. Chapter 6, 59 P( Y 1 = i 1 , . . ., Y k +1 = i k +1 ) =P( Y 1 = i i , . . ., Y k = i k )P( Y k +1 = i k +1 | Y 1 = i 1 , . . ., Y k = i k ) = k !( n − k )! n ! P( n + 1 − k summationdisplay i =1 Y i = i k +1 | Y 1 = i 1 , . . ., Y k = i k ) = braceleftBigg k !( n- k )! n ! , if ∑ k +1 j =1 i j = n + 1 otherwise Thus, the joint mass function is symmetric, which prove the random variables are exchangeable. 2. Chapter 6, 60. The joint mass function is P( X 1 = i 1 , . . ., X n = i n ) = 1 ( n k ) , x i ∈ { , 1 } , i = 1 , . . ., n, n summationdisplay i =1 x i = k which is symmetric in x 1 . . ., x n and the result follows. 3. Chapter 7, 15 Let X i,j , i negationslash = j equal 1 if i and j form a matched pair, and let it be 0 otherwise. Then E( X i,j ) = P( i, j is a matched pair) = 1 n + 1 Hence, the expected number of matched pairs is E( summationdisplay i<j X i,j ) = summationdisplay i,j E( X i,j ) = parenleftbigg n 2 parenrightbigg 1 n ( n −...
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hw6 - Stats 116 HW6 1 Chapter 6 59 P Y 1 = i 1 Y k 1 = i k...

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