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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
Hπββ β¨π₯ β©=π π
π=β¨π₯ β©πβ0
. . .
β¨ πΏπ₯2 β©= β¨π₯ β©π (π₯ )= ππ₯
π₯ !πβπ
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.
π₯T T T T
π=1
π=4π=10
T
π (π₯ )
0.1
0.2
0.3
0.4
HTHT
HT
HT
HT
HT
H
2
Poisson limit
?π=1π=1
β¨π₯ β©=1
π=2π=1 /2
β¨π₯ β©=1 ? ?
? ? ? ?π=4π=1 /4
β¨π₯ β©=1
πβββ¨π₯ β©=π π
π=β¨π₯ β©πβ0
Average total number of heads
T T
3
Poisson limit
HTHT
HT
HT
HT
HT
H
πβββ¨π₯ β©=π π
π=β¨π₯ β©πβ0
H
T T T T T T. . .
. . .
Variance of total number of heads
β¨ πΏπ₯2 β©=ππ (1βπ )β¨π₯ β©π
β¨π₯ β©π
β¨ πΏπ₯2 β©= β¨π₯ β©
T T
4
Poisson limit
HTHT
HT
HT
HT
HT
H
πβββ¨π₯ β©=π π
π=β¨π₯ β©πβ0
H
T T T T T T. . .
. . .
β¨ πΏπ₯2 β©= β¨π₯ β©
Probability distribution for getting x total heads
π (π₯ )= π !(πβπ₯ ) !π₯ !
π π₯ (1βπ )πβπ₯
(ππ₯
π )π₯ (1β ππ₯
π )πβπ₯
(1β ππ₯
π )π (1β ππ₯
π )βπ₯
β1πβππ₯
π (π₯ )= π !(πβπ₯ ) !π₯ ! (ππ₯
π )π₯
πβππ₯
T T
5
Poisson limit
HTHT
HT
HT
HT
HT
H
πβββ¨π₯ β©=π π
π=β¨π₯ β©πβ0
H
T T T T T T. . .
. . .
β¨ πΏπ₯2 β©= β¨π₯ β©
Probability distribution for getting x total heads
π (π₯ )= π !(πβπ₯ ) !π₯ ! (ππ₯
π )π₯
πβππ₯
π β (πβ1 ) β (πβ2 )β― (πβπ₯+1 ) β (πβπ₯ ) !(πβπ₯ )!π₯ !
π (π₯ )=π β (πβ1 ) β (πβ2 )β― (πβπ₯+1 )π₯ ! (ππ₯
π )π₯
πβππ₯
π (π₯ )=ππ₯
π₯
π₯ !πβππ₯
T T T
π (π₯ )
0.1
0.2
0.3
0.4
5 10 15 200
6
HTHT
HT
HT
HT
HT
Hπββ β¨π₯ β©=π π
π=β¨π₯ β©πβ0
. . .
β¨ πΏπ₯2 β©= β¨π₯ β©π (π₯ )= ππ₯
π₯ !πβπ
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.
π₯T T
π=1
π=4π=10
Poisson limit