1. . . . 𝑃 𝑥 = 𝜇 𝑥 𝑥! 𝑒 −𝜇 Poisson limit H 𝑁→∞ 𝑥 =𝑁 𝑝 𝑝= 𝑥 𝑁 →0 𝛿𝑥 2 = 𝑥
𝑃 𝑥 = 𝜇 𝑥 𝑥! 𝑒 −𝜇 H
𝑁→∞
𝑥 =𝑁 𝑝
𝑝= 𝑥 𝑁 →0
𝛿𝑥 2 = 𝑥 H T H T
𝑃 𝑥
0.1
0.2
0.3
0.4
𝜇=1 H T H T H T H T
. . .
𝜇=4
𝜇=10 T T T T T 𝑥 5 10 15 20
Distribution plot adapted from ( licensed by Wikipedia user Skbkekas under a CC-BY-3.0 license.

2. ? ? ? ? ? ? ? Poisson limit 𝑁→∞ 𝑁=1 𝑥 =𝑁 𝑝 𝑝=1 H 𝑝= 𝑥 𝑁 →0 𝑥 =1 𝑁=2
𝑝= 𝑥 𝑁 →0
𝑥 =1
𝑁=2 ? ?
𝑝=1/2
𝑥 =1 H T H T ? ? ? ?
𝑁=4
𝑝=1/4
𝑥 =1 H T H T H T H T

3. . . . . . . Poisson limit 𝑁→∞ H Average total number of heads 𝑥 =𝑁 𝑝
𝑝= 𝑥 𝑁 →0 H T H T
Variance of total number of heads
𝑥 𝑁
𝑥 𝑁 H T H T H T H T
𝛿𝑥 2 = 𝑁 𝑝 1− 𝑝
𝛿𝑥 2 = 𝑥
. . . H
. . . T T T T T T T T

4. . . . . . . Poisson limit 𝑒 − 𝜇 𝑥 𝛿𝑥 2 = 𝑥 𝑁→∞ H 𝑥 =𝑁 𝑝
𝛿𝑥 2 = 𝑥
𝑁→∞ H
𝑥 =𝑁 𝑝
Probability distribution for getting x total heads
𝑝= 𝑥 𝑁 →0 H T H T
𝑃 𝑥 = 𝑁! 𝑁−𝑥 !𝑥! 𝑝 𝑥 1−𝑝 𝑁−𝑥
1− 𝜇 𝑥 𝑁 𝑁−𝑥
𝜇 𝑥 𝑁 𝑥 H T H T H T H T
1− 𝜇 𝑥 𝑁 −𝑥
1− 𝜇 𝑥 𝑁 𝑁 ≈1
𝑒 − 𝜇 𝑥
. . .
𝑃 𝑥 = 𝑁! 𝑁−𝑥 !𝑥! 𝜇 𝑥 𝑁 𝑥 𝑒 − 𝜇 𝑥 H
. . . T T T T T T T T

5. . . . . . . Poisson limit 𝛿𝑥 2 = 𝑥 𝑁→∞ H 𝑥 =𝑁 𝑝
𝛿𝑥 2 = 𝑥
𝑁→∞ H
𝑥 =𝑁 𝑝
Probability distribution for getting x total heads
𝑝= 𝑥 𝑁 →0 H T H T
𝑃 𝑥 = 𝑁! 𝑁−𝑥 !𝑥! 𝜇 𝑥 𝑁 𝑥 𝑒 − 𝜇 𝑥
𝑁∙ 𝑁−1 ∙ 𝑁−2 ⋯ 𝑁−𝑥+1 ∙ 𝑁−𝑥 ! 𝑁−𝑥 !𝑥! H T H T H T H T
𝑃 𝑥 = 𝑁∙ 𝑁−1 ∙ 𝑁−2 ⋯ 𝑁−𝑥+1 𝑥! 𝜇 𝑥 𝑁 𝑥 𝑒 − 𝜇 𝑥
. . .
𝑃 𝑥 = 𝜇 𝑥 𝑥 𝑥! 𝑒 − 𝜇 𝑥 H
. . . T T T T T T T T

6. . . . 𝑃 𝑥 = 𝜇 𝑥 𝑥! 𝑒 −𝜇 Poisson limit H 𝑁→∞ 𝑥 =𝑁 𝑝 𝑝= 𝑥 𝑁 →0 𝛿𝑥 2 = 𝑥
𝑃 𝑥 = 𝜇 𝑥 𝑥! 𝑒 −𝜇 H
𝑁→∞
𝑥 =𝑁 𝑝
𝑝= 𝑥 𝑁 →0
𝛿𝑥 2 = 𝑥 H T H T
𝑃 𝑥
0.1
0.2
0.3
0.4
𝜇=1 H T H T H T H T
. . .
𝜇=4
𝜇=10 T T T T T 𝑥 5 10 15 20
Distribution plot adapted from ( licensed by Wikipedia user Skbkekas under a CC-BY-3.0 license.