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CS70: Satish Rao: Lecture 25. 1. Distribution Comparison 2. Poisson Application 3. Variance
Distributions: continued. Pr [ X = i ] = ( e - 4 ) 4 i i ! E [ X ] = 4 λ = 4 0 1 2 3 4 5 6 7 8 9 10 0 . 1 . 2 Pr [ X = i ] = ( 10 i ) ( . 4 ) i ( . 6 ) 10 - i E [ X ] = 4 n = 10 , p = 4 0 1 2 3 4 5 6 7 8 9 10 0 . 1 . 2 p = . 25 Pr [ X = i ] = . 75 i - 1 × . 25 E [ X ] = 4 0 1 2 3 4 5 6 7 8 9 10 0 . 1 . 2

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Poisson and binomial.. Poisson: “balls in bin 1 when n balls into n / λ bins.” Binomial: n , p = λ n . In the limit of large n , Binomial converges to Poisson.
Poisson and binomial.. Poisson: “balls in bin 1 when n balls into n / λ bins.” Binomial: n , p = λ n . In the limit of large n , Binomial converges to Poisson.

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How big a server? 1. Say 14,400 requests arrive in an hour? 2. Each request takes a second to process. How many servers should you have to keep up?
How big a server? 1. Say 14,400 requests arrive in an hour? 2. Each request takes a second to process. How many servers should you have to keep up? 14400 / 3600 = 4

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How big a server? 1. Say 14,400 requests arrive in an hour? 2. Each request takes a second to process. How many servers should you have to keep up? 14400 / 3600 = 4 So four servers?
Modeling: How big a server? 1. n customers, n is unknown. 2. Each makes request with (small) probability p . (Rare Event.) Don’t know n and p .

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Modeling: How big a server? 1. n customers, n is unknown. 2. Each makes request with (small) probability p . (Rare Event.) Don’t know n and p . Can’t use binomial.
Modeling: How big a server? 1. n customers, n is unknown. 2. Each makes request with (small) probability p . (Rare Event.) Don’t know n and p . Can’t use binomial. Use Poisson!

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Modeling: How big a server? 1. n customers, n is unknown. 2. Each makes request with (small) probability p . (Rare Event.) Don’t know n and p . Can’t use binomial. Use Poisson! With λ = 14 , 400 / 3 , 600 =4.
Poisson and servers. Pr [ X = i ] = ( e - 4 ) 4 i i ! E [ X ] = 4 λ = 4 0 1 2 3 4 5 6 7 8 9 10 0 . 1 . 2

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Poisson and servers. Pr [ X = i ] = ( e - 4 ) 4 i i ! E [ X ] = 4 λ = 4 0 1 2 3 4 5 6 7 8 9 10 0 . 1 . 2 Can I usually keep up with the load?
Poisson and servers. Pr [ X = i ] = ( e - 4 ) 4 i i ! E [ X ] = 4 λ = 4 0 1 2 3 4 5 6 7 8 9 10 0 . 1 . 2 Can I usually keep up with the load? What is Pr [ X “number of servers” ] ?

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Poisson and servers. Pr [ X = i ] = ( e - 4 ) 4 i i ! E [ X ] = 4 λ = 4 0 1 2 3 4 5 6 7 8 9 10 0 . 1 . 2 Can I usually keep up with the load? What is Pr [ X “number of servers” ] ? i Pr [ X i ] 4 . 628
Poisson and servers. Pr [ X = i ] = ( e - 4 ) 4 i i ! E [ X ] = 4 λ = 4 0 1 2 3 4 5 6 7 8 9 10 0 . 1 . 2 Can I usually keep up with the load? What is Pr [ X “number of servers” ] ? i Pr [ X i ] 4 . 628 6 . 888

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Poisson and servers. Pr [ X = i ] = ( e - 4 ) 4 i i ! E [ X ] = 4 λ = 4 0 1 2 3 4 5 6 7 8 9 10 0 . 1 . 2 Can I usually keep up with the load? What is Pr [ X “number of servers” ] ? i Pr [ X i ] 4 . 628 6 . 888 8 . 978
Poisson and servers. Pr [ X = i ] = ( e - 4 ) 4 i i ! E [ X ] = 4 λ = 4 0 1 2 3 4 5 6 7 8 9 10 0 . 1 . 2 Can I usually keep up with the load? What is Pr [ X “number of servers” ] ? i Pr [ X i ] 4 . 628 6 . 888 8 . 978 10 . 996

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Poisson and servers.
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