This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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...
View
Full
Document
This note was uploaded on 11/15/2008 for the course STATS 116 taught by Professor Staff during the Summer '07 term at Stanford.
 Summer '07
 Staff
 Probability

Click to edit the document details