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Poisson limit. 0.4. H. H. H. H. H. H. H. T. T. T. T. T. T. T. T. T. T. 0.3. 0.2. 0.1. T. 0. 5. 10. 15. 20. Distribution plot adapted from (http :// en.wikipedia.org/wiki/File:Poisson_pmf.svg), licensed by Wikipedia user Skbkekas under a CC-BY-3.0 license. - PowerPoint PPT Presentation
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5 10 15 200
1
Poisson limit
HTHT
HT
HT
HT
HT
H𝑁→∞ ⟨𝑥 ⟩=𝑁 𝑝
𝑝=⟨𝑥 ⟩𝑁→0
. . .
⟨ 𝛿𝑥2 ⟩= ⟨𝑥 ⟩𝑃 (𝑥 )= 𝜇𝑥
𝑥 !𝑒−𝜇
Distribution plot adapted from (http://en.wikipedia.org/wiki/File:Poisson_pmf.svg), licensed by Wikipedia user Skbkekas under a CC-BY-3.0 license.
𝑥T T T T
𝜇=1
𝜇=4𝜇=10
T
𝑃 (𝑥 )
0.1
0.2
0.3
0.4
HTHT
HT
HT
HT
HT
H
2
Poisson limit
?𝑁=1𝑝=1
⟨𝑥 ⟩=1
𝑁=2𝑝=1 /2
⟨𝑥 ⟩=1 ? ?
? ? ? ?𝑁=4𝑝=1 /4
⟨𝑥 ⟩=1
𝑁→∞⟨𝑥 ⟩=𝑁 𝑝
𝑝=⟨𝑥 ⟩𝑁→0
Average total number of heads
T T
3
Poisson limit
HTHT
HT
HT
HT
HT
H
𝑁→∞⟨𝑥 ⟩=𝑁 𝑝
𝑝=⟨𝑥 ⟩𝑁→0
H
T T T T T T. . .
. . .
Variance of total number of heads
⟨ 𝛿𝑥2 ⟩=𝑁𝑝 (1−𝑝 )⟨𝑥 ⟩𝑁
⟨𝑥 ⟩𝑁
⟨ 𝛿𝑥2 ⟩= ⟨𝑥 ⟩
T T
4
Poisson limit
HTHT
HT
HT
HT
HT
H
𝑁→∞⟨𝑥 ⟩=𝑁 𝑝
𝑝=⟨𝑥 ⟩𝑁→0
H
T T T T T T. . .
. . .
⟨ 𝛿𝑥2 ⟩= ⟨𝑥 ⟩
Probability distribution for getting x total heads
𝑃 (𝑥 )= 𝑁 !(𝑁−𝑥 ) !𝑥 !
𝑝 𝑥 (1−𝑝 )𝑁−𝑥
(𝜇𝑥
𝑁 )𝑥 (1− 𝜇𝑥
𝑁 )𝑁−𝑥
(1− 𝜇𝑥
𝑁 )𝑁 (1− 𝜇𝑥
𝑁 )−𝑥
≈1𝑒−𝜇𝑥
𝑃 (𝑥 )= 𝑁 !(𝑁−𝑥 ) !𝑥 ! (𝜇𝑥
𝑁 )𝑥
𝑒−𝜇𝑥
T T
5
Poisson limit
HTHT
HT
HT
HT
HT
H
𝑁→∞⟨𝑥 ⟩=𝑁 𝑝
𝑝=⟨𝑥 ⟩𝑁→0
H
T T T T T T. . .
. . .
⟨ 𝛿𝑥2 ⟩= ⟨𝑥 ⟩
Probability distribution for getting x total heads
𝑃 (𝑥 )= 𝑁 !(𝑁−𝑥 ) !𝑥 ! (𝜇𝑥
𝑁 )𝑥
𝑒−𝜇𝑥
𝑁 ∙ (𝑁−1 ) ∙ (𝑁−2 )⋯ (𝑁−𝑥+1 ) ∙ (𝑁−𝑥 ) !(𝑁−𝑥 )!𝑥 !
𝑃 (𝑥 )=𝑁 ∙ (𝑁−1 ) ∙ (𝑁−2 )⋯ (𝑁−𝑥+1 )𝑥 ! (𝜇𝑥
𝑁 )𝑥
𝑒−𝜇𝑥
𝑃 (𝑥 )=𝜇𝑥
𝑥
𝑥 !𝑒−𝜇𝑥
T T T
𝑃 (𝑥 )
0.1
0.2
0.3
0.4
5 10 15 200
6
HTHT
HT
HT
HT
HT
H𝑁→∞ ⟨𝑥 ⟩=𝑁 𝑝
𝑝=⟨𝑥 ⟩𝑁→0
. . .
⟨ 𝛿𝑥2 ⟩= ⟨𝑥 ⟩𝑃 (𝑥 )= 𝜇𝑥
𝑥 !𝑒−𝜇
Distribution plot adapted from (http://en.wikipedia.org/wiki/File:Poisson_pmf.svg), licensed by Wikipedia user Skbkekas under a CC-BY-3.0 license.
𝑥T T
𝜇=1
𝜇=4𝜇=10
Poisson limit