# ANS_HW4 - 550.310 HW4 Solutions 4.5 4.6 4.7 4.8 4.12 4.17...

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550.310 HW4 Solutions Please do NOT redistribute!!! 4.5, 4.6, 4.7, 4.8, 4.12, 4.17*, 4.19*, 4.22* It’s a binomial distribution: 23 32 4 555 { 2 }{ 3 4 } ( 1) ( ( 234 PPX PX p p p p p p ⎛⎞ == += = + + ⎜⎟ ⎝⎠ 1 where Thus . 1/4. p = 0.3662 P = 4.6 It’s a binomial distribution: 35 8 { 3} (1 ) 0.2787 3 p p = = 4.7 It’s a binomial distribution: 1 { 1} 0.5 1 { 0} 0.5 { 0.5 ) 0.5 log 0.5 13.51 n p p n ≥≥ ⇒− = ≥ ⇒ = ≤ ⇒− ≤ ⇒≥ = Be aware that 1 log p x is a decreasing function because 11 p < . 4.8 07 16 77 { { ) ) 0.9556 01 p p p p = + = Comparing to 4 0.95 0.8145 = 4.12 0 l 0,0 0,0.5 1( rP 0 ) = = 1 l 1,0 1,0.5 12 (0) = = , 1,1 1,0.5 ) = = 2 l 2,0 2,0.5 14 , 2,1 2,0.5 ) = = , 2,2 2,0.5 (2) = = It is all satisfied for n = 1, 2, 3. For the sake of induction, assume ,, 0 . 5 () nk n k = . 1 1 1 1 1, , 1 , ,0.5 ,0.5 22 2 2 2 2 1 1,0.5 1 ) ( ) ( ) ( ) 1 1 ()() nn n k n n kn k n n rrr P k P k kk k n Pk k ++ +− + =+ = + = + = + +

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4.17 0 1000 1 999 2 998 3 997 1 (0) (1) (2) (3) 1000 1000 1000 1000 1 (1 ) ) ) ) 01 2 3 PPP PP p pp p p p p = −− −−= ⎛⎞ ⎜⎟ ⎝⎠ p use Poisson approximation 6.7 np λ == 0123 () 1 0.9012 0! 1! 2! 3! np np np np np np np np Pe e e e −−−− =− = 4.19 use Poisson approximation 1.2 np 1 0.0338 1! 3! np np np np np np np np e e e = 4.22 2 (2 )! { 2 : 0,1,2,. ..} ( ) i s s i PX ii e
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## This note was uploaded on 10/21/2008 for the course MATH 302 taught by Professor Goldberg during the Fall '08 term at Johns Hopkins.

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ANS_HW4 - 550.310 HW4 Solutions 4.5 4.6 4.7 4.8 4.12 4.17...

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