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Unformatted text preview: ∞ eλx dx = 1 λ 4. For α > 1, E [ Y ] = ∫ ∞ 1 y α y α +1 dy = ∫ ∞ 1 αyα dy =α α1 y( α1)  ∞ 1 = α α1 5. (a) Since LHS = k n ! ( nk )! k ! = n ! ( nk )!( k1)! 1 ISyE 2027B Probability with Applications Fall 2010 RHS = n ( n1)! ( n1( k1))!( k1)! = n ! ( nk )!( k1)! So LHS = RHS. Done. (b) E [ Z ] = n ∑ k =0 k ( n k ) p k (1p ) nk = n ∑ k =1 k ( n k ) p k (1p ) nk = n ∑ k =1 n ( n1 k1 ) pp k1 (1p ) nk by (a) = np n ∑ k =1 ( n1 k1 ) p k1 (1p ) nk = np n1 ∑ j =0 ( n1 j ) p j (1p ) ( n1)j let j = k1 = np ( p + (1p )) n1 by the binomial theorem* = np You can also see that as the sum of a binomial probability mass function. 2...
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This note was uploaded on 02/21/2011 for the course ISYE 2027 taught by Professor Zahrn during the Fall '08 term at Georgia Tech.
 Fall '08
 Zahrn

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