poisson limit

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