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hw06_sol

# hw06_sol - Homework 6 Problem 1 Let Î be a rate parameter...

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Unformatted text preview: Homework 6 October 17, 2007 Problem 1. Let Î» be a rate parameter. Then notice that P ( X > . 01) = 1- P ( X â‰¤ . 01) = e- . 01 Î» . Solving 1 / 2 = e- . 01 Î» , we have Î» = 100ln2 . Thus, P ( X > t ) = e- 100 t ln2 , so by solving . 9 = e- 100 t ln2 , we have t =- . 01log 2 . 9 . Problem 2. EX = R âˆž-âˆž x Â· Î»e- Î» | x | dx = R-âˆž 1 2 xÎ»e Î»x + R âˆž 1 2 xÎ»e- Î»x = 1 2 R-âˆž xde Î»x- 1 2 R âˆž xde- Î»x =- 1 2 R-âˆž e Î»x dx + 1 2 R âˆž e- Î»x dx =- 1 2 Î» + 1 2 Î» = 0 E [ X 2 ] = R âˆž-âˆž x 2 Â· Î»e- Î» | x | dx = R-âˆž 1 2 x 2 Î»e Î»x + R âˆž 1 2 x 2 Î»e- Î»x = 1 2 R-âˆž x 2 de Î»x- 1 2 R âˆž x 2 de- Î»x =- R-âˆž xe Î»x dx + R âˆž xe- Î»x dx = 1 Î» 2 + 1 Î» 2 = 2 Î» 2 V arX = E [ X 2 ]- ( EX ) 2 = 2 Î» 2 Problem 3. If X âˆ¼ Bin ( n,p ) , then Ï† X ( t ) = Ee tX = n X j =0 P [ X = j ] = n X j =0 e tj p j (1- p ) n- j n j = n X j =0 ( e t p ) j (1- p ) n- j n j = ( e t p + (1- p ) ) n , 1 where the last step invokes the binomial theorem. So Ï† X ( t ) = npe t ( e t p + (1- p )...
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hw06_sol - Homework 6 Problem 1 Let Î be a rate parameter...

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