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Unformatted text preview: Math Brushup ST520  Hannig Adam Labadorf ST520  Hannig August 27, 2007 Math Brushup 1. ∞ X i =1 ∞ X j = i p j + i The result of this equation depends on the value of p . If p < , p 6 = 1 or p > 1 the equation will always diverge. If p = 1 the value is unknown as the equation will oscillate between 0 and 1 infinitely. If ≤ p ≤ 1 the value can be found in closed form as follows: ∞ X i =1 ∞ X j = i p i p j ∞ X i =1 p i ∞ X j = i p j (1) Since ≤ p ≤ 1 , we can apply the closed form in Eq. (2): ∞ X k = m ar k = ar m 1 r (2) Our equation becomes: ∞ X i =1 p i p i 1 p 1 1 p ∞ X i =1 p i p i 1 1 p ∞ X i =1 ( p 2 ) i Again applying (2): 1 1 p p 2 1 p 2 (3) 2. Z 2 1 1 x dx = Z 2 1 x 1 dx ln (  x  ) 2 x =1 = ln (  x  )  x =2 ln (  x  )  x =1 = 0 . 6931 1 Math Brushup ST520  Hannig 3. Z 1 1 1 p  x  Z 1 1 p  x  + Z 1 1 p  x  = 2 Z 1 1 p  x  2 Z 1 (  x  . 5 ) 1 = 2 Z 1  x  . 5 2  x  . 5...
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This note was uploaded on 06/16/2009 for the course STAT 520 taught by Professor Hannig,j. during the Spring '09 term at Colorado State.
 Spring '09
 Hannig,J.
 Statistics, Probability

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