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7/23/2019 Solution of Stochastic Process and Modelling
http://slidepdf.com/reader/full/solution-of-stochastic-process-and-modelling 1/31
❙ ♦ ❝ ❤ ❛ ✐ ❝ ♦ ❝ ❡ ❡ ❛ ♥ ❞ ▼ ♦ ❞ ❡ ❧ ✐ ♥ ❣ ✭ ▼ ❆ ❚ ❍ ✻ ✻ ✻ ✵ ✮
❆ ✐ ❣ ♥ ♠ ❡ ♥ ✸
● ♦ ✉ ❛ ✈ ❙ ❛ ❤ ❛ ✭ ❘ ■ ◆ ✿ ✻ ✻ ✶ ✺ ✸ ✾ ✽ ✻ ✻ ✮
✶ ❆ ♣ ♣ ❧ ✐ ❝ ❛ ✐ ♦ ♥ ❛ ♥ ❞ ❈ ❛ ❧ ❝ ✉ ❧ ❛ ✐ ♦ ♥ ♦ ❜ ❧ ❡ ♠
✶ ✳ ✶ ❘ ❛ ✐ ♥ ❛ ▼ ❛ ❦ ♦ ✈ ▼ ❛ ③ ❡
✶ ✷ ✸ ✹
✺ ✻ ✼ ✽
✾ ✶ ✵ ✶ ✶ ✶ ✷
✶ ✸ ✶ ✹ ✶ ✺ ✶ ✻
❚ ❛ ❜ ❧ ❡ ✶ ✿ ❋ ✐ ❣ ✉ ❡ ❤ ♦ ✇ ✐ ♥ ❣ ❤ ❡ ♠ ❛ ③ ❡ ❛ ❧ ♦ ♥ ❣ ✇ ✐ ❤ ✐ ♥ ❞ ✐ ❝ ❡ ❡ ♣ ❡ ❡ ♥ ✐ ♥ ❣ ❤ ❡ ❛ ❡ ✳ ❘ ❛ ❤ ❜ ❡ ❛ ❢ ♦ ♠ ❙ ❛ ❡ ✲ ✶ ✻ ✱ ❤ ❡ ❤ ♦ ❝ ❦ ✐
❧ ♦ ❝ ❛ ❡ ❞ ✐ ♥ ❙ ❛ ❡ ✲ ✽ ❛ ♥ ❞ ❙ ❛ ❡ ✲ ✾ ✱ ❤ ❡ ❝ ❤ ❡ ❡ ❡ ✐ ❧ ♦ ❝ ❛ ❡ ❞ ✐ ♥ ❙ ❛ ❡ ✲ ✸ ❛ ♥ ❞ ❙ ❛ ♠ ❛ ❢ ♦ ♠ ❙ ❛ ❡ ✲ ✶ ✳
✭ ❛ ✮ ❚ ❤ ❡ ✇ ❛ ② ❘ ❛ ❤ ❜ ❡ ♠ ♦ ✈ ❡ ✐ ♥ ❤ ❡ ♠ ❛ ③ ❡ ✐ ❣ ✉ ✐ ❞ ❡ ❞ ❜ ② ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ❤ ❡ ❡ ✉ ❧ ❡ ✿
• ❘ ✉ ❧ ❡ ✲ ✶ ✿ ■ ♥ ❡ ✈ ❡ ② ❡ ♣ ♦ ❝ ❤ ❤ ❡ ❡ ✐ ❛ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② prest = 0.1 ❤ ❛ ❘ ❛ ❤ ❜ ❡ ✇ ♦ ♥ ✬ ♠ ♦ ✈ ❡ ❜ ❡ ❝ ❛ ✉ ❡ ❤ ❡ ✴ ❤ ❡ ✐ ✐ ❡ ❞ ❛ ♥ ❞
✇ ❛ ♥ ♦ ❡ ✳
• ❘ ✉ ❧ ❡ ✲ ✷ ✿ ■ ♥ ❡ ✈ ❡ ② ❡ ♣ ♦ ❝ ❤ ✱ ❘ ❛ ❤ ❜ ❡ ❝ ❛ ♥ ♠ ♦ ✈ ❡ ♦ ♥ ❧ ② ♦ ♥ ❡ ❡ ♣ ❡ ✐ ❤ ❡ ✉ ♣ ♦ ❞ ♦ ✇ ♥ ♦ ❧ ❡ ❢ ♦ ✐ ❣ ❤ ♣ ♦ ✈ ✐ ❞ ❡ ❞ ❤ ❛ ❤ ❡ ❡ ✐
♥ ♦ ♦ ❜ ❛ ❝ ❧ ❡ ✐ ♥ ❤ ❛ ❞ ✐ ❡ ❝ ✐ ♦ ♥ ✳ ❉ ✐ ❛ ❣ ♦ ♥ ❛ ❧ ♠ ♦ ✈ ❡ ♠ ❡ ♥ ✐ ◆ ❖ ❚ ❛ ❧ ❧ ♦ ✇ ❡ ❞ ✳
• ❘ ✉ ❧ ❡ ✲ ✸ ✿ ❙ ❛ ② ❤ ❛ ✐ ♥ ❛ ❣ ✐ ✈ ❡ ♥ ❛ ❡ ❘ ❛ ❤ ❜ ❡ ❝ ❛ ♥ ♠ ♦ ✈ ❡ ✐ ♥ m ∈ {1, 2, 3, 4} ♣ ♦ ✐ ❜ ❧ ❡ ❞ ✐ ❡ ❝ ✐ ♦ ♥ ✳ ❚ ❤ ❡ ♥ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ②
❤ ❛ ✐ ✇ ✐ ❧ ❧ ♠ ♦ ✈ ❡ ✐ ♥ ♦ ♥ ❡ ♦ ❢ ❤ ❡ ❡ ❞ ✐ ❡ ❝ ✐ ♦ ♥ ✐
1− prestm
= 0.9m
✱ ✐ ✳ ❡ ✳ ❘ ❛ ❤ ❜ ❡ ❤ ❛ ❡ ✉ ❛ ❧ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ♦ ♠ ♦ ✈ ❡ ✐ ♥ ❛ ♥ ②
❞ ✐ ❡ ❝ ✐ ♦ ♥ ✳
■ ▼ ❖ ❘ ❚ ❆ ◆ ❚ ✿ ❲ ✐ ❤ ♦ ✉ ❘ ✉ ❧ ❡ ✲ ✶ ❤ ❡ ♠ ❛ ❦ ♦ ✈ ❝ ❤ ❛ ✐ ♥ ✇ ✐ ❧ ❧ ❜ ❡ ♣ ❡ ✐ ♦ ❞ ✐ ❝ ✇ ✐ ❤ ♣ ❡ ✐ ♦ ❞ 2✳ ❚ ❤ ✐ ♠ ❛ ② ❧ ❡ ❛ ❞ ♦ ❝ ♦ ♠ ♣ ❧ ✐ ❝ ❛ ✐ ♦ ♥ ✐ ♥
❝ ♦ ♠ ♣ ✉ ❛ ✐ ♦ ♥ ✳ ❍ ❡ ♥ ❝ ❡ ✇ ❡ ✐ ♠ ♣ ♦ ❡ ❞ ❘ ✉ ❧ ❡ ✲ ✶ ♦ ♠ ❛ ❦ ❡ ❤ ❡ ♠ ❛ ❦ ♦ ✈ ❝ ❤ ❛ ✐ ♥ ❛ ♣ ❡ ✐ ♦ ❞ ✐ ❝ ✳
❚ ❤ ❡ ❛ ❜ ♦ ✈ ❡ ✉ ❧ ❡ ❧ ❡ ❛ ❞ ♦ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ① P ✳
✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶ ✵ ✶ ✶ ✶ ✷ ✶ ✸ ✶ ✹ ✶ ✺ ✶ ✻
✶ ✵ ✳ ✶ ✵ ✵ ✵ ✵ ✳ ✾ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵
✷ ✵ ✵ ✳ ✶ ✵ ✳ ✹ ✺ ✵ ✵ ✵ ✳ ✹ ✺ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵
✸ ✵ ✵ ✳ ✹ ✺ ✵ ✳ ✶ ✵ ✳ ✹ ✺ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵
✹ ✵ ✵ ✵ ✳ ✹ ✺ ✵ ✳ ✶ ✵ ✵ ✵ ✵ ✳ ✹ ✺ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵
✺ ✵ ✳ ✹ ✺ ✵ ✵ ✵ ✵ ✳ ✶ ✵ ✳ ✹ ✺ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵
✻ ✵ ✵ ✳ ✷ ✷ ✺ ✵ ✵ ✵ ✳ ✷ ✷ ✺ ✵ ✳ ✶ ✵ ✳ ✷ ✷ ✺ ✵ ✵ ✵ ✳ ✷ ✷ ✺ ✵ ✵ ✵ ✵ ✵ ✵
✼ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✹ ✺ ✵ ✳ ✶ ✵ ✳ ✹ ✺ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵
✽ ✵ ✵ ✵ ✵ ✳ ✹ ✺ ✵ ✵ ✵ ✳ ✹ ✺ ✵ ✳ ✶ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵
✾ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✶ ✵ ✵ ✵ ✵ ✳ ✾ ✵ ✵ ✵
✶ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✹ ✺ ✵ ✵ ✵ ✵ ✳ ✶ ✵ ✳ ✹ ✺ ✵ ✵ ✵ ✵ ✵
✶ ✶ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✹ ✺ ✵ ✳ ✶ ✵ ✵ ✵ ✵ ✳ ✹ ✺ ✵
✶ ✷ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✶ ✵ ✵ ✵ ✵ ✳ ✾
✶ ✸ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✹ ✺ ✵ ✵ ✵ ✵ ✳ ✶ ✵ ✳ ✹ ✺ ✵ ✵
✶ ✹ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✹ ✺ ✵ ✳ ✶ ✵ ✳ ✹ ✺ ✵
✶ ✺ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✸ ✵ ✵ ✳ ✸ ✵ ✳ ✶ ✵ ✳ ✸
✶ ✻ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✹ ✺ ✵ ✵ ✵ ✳ ✹ ✺ ✵ ✳ ✶
❚ ❛ ❜ ❧ ❡ ✷ ✿ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ① P ❣ ♦ ✈ ❡ ♥ ✐ ♥ ❣ ❤ ❡ ♠ ♦ ✈ ❡ ♠ ❡ ♥ ♦ ❢ ❘ ❛ ❤ ❜ ❡ ✐ ♥ ❤ ❡ ♠ ❛ ③ ❡ ✳
✶
7/23/2019 Solution of Stochastic Process and Modelling
http://slidepdf.com/reader/full/solution-of-stochastic-process-and-modelling 2/31
▲ ❊ ❆ ❙ ❊ ◆ ❖ ❚ ❊ ✿
✶ ✮ ❚ ❤ ❡ ▼ ❛ ❦ ♦ ✈ ❈ ❤ ❛ ✐ ♥ ❣ ♦ ✈ ❡ ♥ ❡ ❞ ❜ ② ❤ ❡ ❛ ❜ ♦ ✈ ❡ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ① ✐ ✐ ❡ ❞ ✉ ❝ ✐ ❜ ❧ ❡ ❛ ♥ ❞ ❢ ♦ ♠ ❛ ❝ ❧ ♦ ❡ ❞ ❝ ♦ ♠ ♠ ✉ ♥ ✐ ❝ ❛ ✲
✐ ♦ ♥ ❝ ❧ ❛ ✳
✷ ✮ ❙ ✐ ♥ ❝ ❡ ❤ ❡ ▼ ❛ ❦ ♦ ✈ ❈ ❤ ❛ ✐ ♥ ❢ ♦ ♠ ❛ ❝ ❧ ♦ ❡ ❞ ❝ ♦ ♠ ♠ ✉ ♥ ✐ ❝ ❛ ✐ ♦ ♥ ❝ ❧ ❛ ✱ ✇ ❡ ❝ ❛ ♥ ✉ ❡ ❤ ❡ ❝ ♦ ♥ ❝ ❡ ♣ ❢ ♦ ♠ ▲ ❡ ❝ ✉ ❡ ✲ ✶ ✷ ✭ 19th ❖ ❝ ♦ ❜ ❡ ✱
✷ ✵ ✶ ✺ ✮ ♦ ❛ ♥ ✇ ❡ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ♣ ❛ ✳ ❚ ♦ ❛ ♥ ✇ ❡ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ♣ ❛ ✇ ❡ ♠ ✐ ❣ ❤ ♥ ♦ ❣ ✐ ✈ ❡ ❛ ❞ ❡ ❛ ✐ ❧ ❡ ❞ ❡ ① ♣ ❧ ❛ ♥ ❛ ✐ ♦ ♥ ♦ ❢ ❤ ❡ ❢ ♦ ♠ ✉ ❧ ❛
✉ ❡ ❞ ✳ ❘ ❛ ❤ ❡ ✱ ✇ ❡ ✇ ✐ ❧ ❧ ❥ ✉ ♠ ❛ ♣ ❤ ❡ ✉ ❡ ✐ ♦ ♥ ♦ ✇ ❤ ❛ ✇ ❡ ❧ ❡ ❛ ♥ ❡ ❞ ✐ ♥ ▲ ❡ ❝ ✉ ❡ ✲ ✶ ✷ ❛ ♥ ❞ ❞ ✐ ❡ ❝ ❧ ② ✉ ❡ ❤ ❡ ❢ ♦ ♠ ✉ ❧ ❛ ❢ ♦ ♠ ▲ ❡ ❝ ✉ ❡ ✲ ✶ ✷ ✳
✸ ✮ ❲ ❡ ♠ ✐ ❣ ❤ ❤ ❛ ✈ ❡ ❞ ♦ ♥ ❡ ♦ ♠ ❡ ♥ ♦ ❛ ✐ ♦ ♥ ♦ ✈ ❡ ❧ ♦ ❛ ❞ ✐ ♥ ❣
✶
♦ ❛ ♥ ✇ ❡ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ♣ ❛ ♦ ❢ ❤ ❡ ✉ ❡ ✐ ♦ ♥ ✳
✹ ✮ ❚ ❤ ❡ ▼ ❆ ❚ ▲ ❆ ❇ ❝ ♦ ❞ ❡ ❝ ♦ ❡ ♣ ♦ ♥ ❞ ✐ ♥ ❣ ♦ ❛ ❧ ❧ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ♣ ❛ ✐ ✐ ♥ ❝ ❧ ✉ ❞ ❡ ❞ ✐ ♥ ❆ ♣ ♣ ❡ ♥ ❞ ✐ ① ✲ ❆ ✳
✭ ❜ ✮ ❚ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ❝ ❛ ♥ ❜ ❡ ❛ ❜ ❛ ❝ ❡ ❞ ❛ ✿ ❙ ❛ ✐ ♥ ❣ ❢ ♦ ♠ ❛ ❡ i✱ ✇ ❤ ❛ ✐ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❡ ♣ ♦ ❝ ❤ ✉ ♥ ✐ ❧ ❛ ❡ j ✐
❡ ❛ ❝ ❤ ❡ ❞ ❄ ■ ♥ ❤ ✐ ✉ ❡ ✐ ♦ ♥ i = 16 ❛ ♥ ❞ j = 3✳ ❚ ♦ ❛ ♥ ✇ ❡ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ✇ ❡ ❛ ❞ ♦ ♣ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ❡ ♣ ✿
❙ ❡ ♣ ✲ ✶ ✿ ❲ ❡ ♠ ❛ ❦ ❡ ❛ ❡ j = 3 ❛ ❜ ♦ ❜ ✐ ♥ ❣ ✳ ❚ ♦ ❞ ♦ ❤ ✐ ✇ ❡ ♠ ❛ ❦ ❡ ❛ ♠ ♦ ❞ ✐ ❢ ❡ ❞ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ①
P ❛ ❢ ♦ ❧ ❧ ♦ ✇ ✿ ✶ ✮
❙ ❡
P = P ✳ ✷ ✮ ❙ ❡ ❛ ❧ ❧ ❤ ❡ ❡ ❧ ❡ ♠ ❡ ♥ ♦ ❢ ❤ ❡ 3rd ♦ ✇ ♦ ❢
P ❛ 0 ❡ ① ❝ ❡ ♣ p33 ✇ ❤ ✐ ❝ ❤ ✐ ❡ ♦ 1 ✳
❙ ❡ ♣ ✲ ✷ ✿ ❲ ❡ ❝ ♦ ♥ ✉ ❝ ❛ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ① Q ✭ ❛ ✐ ❝ ❧ ② ✉ ❜ ✲ ♦ ❝ ❤ ❛ ✐ ❝ ♠ ❛ ✐ ① ✮ ✇ ❤ ✐ ❝ ❤ ❝ ♦ ♥ ❛ ✐ ♥ ❤ ❡ ♦ ♥ ❡ ✲ ❡ ♣
❛ ♥ ✐ ✐ ♦ ♥ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❢ ♦ ♠ ♦ ♥ ❡ ❛ ♥ ✐ ❡ ♥ ❛ ❡ ♦ ❛ ♥ ♦ ❤ ❡ ❛ ♥ ✐ ❡ ♥ ❛ ❡ ✳ ■ ♥ ❤ ❡ ♠ ♦ ❞ ✐ ✜ ❡ ❞ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ①
P ✱ ❛ ❧ ❧ ❛ ❡ ❛ ❡ ❛ ♥ ✐ ❡ ♥ ❡ ① ❝ ❡ ♣ ❙ ❛ ❡ ✲ ✸ ✳ ❍ ❡ ♥ ❝ ❡ ✇ ❡ ❞ ❡ ❧ ❡ ❡ ❤ ❡
3rd
♦ ✇ ❛ ♥ ❞
3rd
❝ ♦ ❧ ✉ ♠ ♥ ❢ ♦ ♠ P ♦ ❣ ❡
Q✳
❙ ❡ ♣ ✲ ✸ ✿ ▲ ❡ ❛ ② ❤ ❛ ❤ ❡ ✐ ♠ ❡ ❝ ♦ ❡ ♣ ♦ ♥ ❞ ✐ ♥ ❣ ♦ ❡ ❛ ❝ ❤ ❡ ♣ ♦ ❝ ❤ ✐ 1 ✳ ❲ ❡ ❤ ❡ ❡ ❢ ♦ ❡ ❡ ❤ ❡ ❛ ❞ ❞ ✐ ✐ ✈ ❡ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ❛ ❧ f ≡ 1 ✳ ▼ ♦ ❡
♣ ❡ ❝ ✐ ❡ ❧ ② ✇ ❡ ❞ ❡ ✜ ♥ ❡ ❛ ❝ ♦ ❧ ✉ ♠ ♥ ✈ ❡ ❝ ♦ f ∈ R15❛ ♥ ❞ ❡ ❛ ❧ ❧ ✐ ❡ ❧ ❡ ♠ ❡ ♥ ❛ 1 ✳
❙ ❡ ♣ ✲ ✹ ✿ ❲ ❡ ❝ ♦ ♠ ♣ ✉ ❡ ❤ ❡ ✈ ❡ ❝ ♦
µτ = (I − Q)−1
f
✇ ❤ ✐ ❝ ❤ ❤ ♦ ✇ ❤ ❡ ❛ ✈ ❡ ❛ ❣ ❡ ✐ ♠ ❡ ♦ ❡ ❛ ❝ ❤ ❤ ❡ ❝ ❤ ❡ ❡ ❡ ❛ ✐ ♥ ❣ ❢ ♦ ♠ ❛ ♥ ② ❛ ♥ ✐ ❡ ♥ ❛ ❡ ✳
❙ ❡ ♣ ✲ ✺ ✿ ❲ ❡ ❛ ❡ ✐ ♥ ❡ ❡ ❡ ❞ ✐ ♥ ❤ ❡ ❛ ✈ ❡ ❛ ❣ ❡ ✐ ♠ ❡ ♦ ❡ ❛ ❝ ❤ ❤ ❡ ❝ ❤ ❡ ❡ ❡ ❣ ✐ ✈ ❡ ♥ ❤ ❛ ❘ ❛ ❤ ❜ ❡ ❛ ❢ ♦ ♠ ❙ ❛ ❡ ✲ ✶ ✻ ✳ ❚ ❤ ✐ ✐ ❣ ✐ ✈ ❡ ♥
❜ ② µτ (15)✱ ❤ ❡ 15th ♦ ✇ ♦ ❢ µτ ✳
❚ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ✐ ♠ ❡ ✐ ❢ ♦ ✉ ♥ ❞ ♦ ❜ ❡ 85.1852 ✳
✭ ❝ ✮ ❚ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ❝ ❛ ♥ ❜ ❡ ❛ ❜ ❛ ❝ ❡ ❞ ❛ ✿ ❙ ❛ ✐ ♥ ❣ ❢ ♦ ♠ ❛ ❡ i✱ ✇ ❤ ❛ ✐ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❡ ♣ ♦ ❝ ❤ ♣ ❡ ♥ ✐ ♥ ❛ ❡ k ✭ k
❝ ❛ ♥ ❜ ❡ ❛ ✈ ❡ ❝ ♦ ❝ ♦ ♥ ✐ ✐ ♥ ❣ ♦ ❢ ♠ ❛ ♥ ② ❛ ❡ ✮ ✱ ❜ ❡ ❢ ♦ ❡ ❛ ❡ j ✐ ✈ ✐ ✐ ❡ ❞ ❄ ■ ♥ ❤ ✐ ✉ ❡ ✐ ♦ ♥ i = 16 ✱ k =
8 9
❛ ♥ ❞ j = 3 ✳ ❚ ♦
❛ ♥ ✇ ❡ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ✇ ❡ ❛ ❞ ♦ ♣ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ❡ ♣ ✿
❙ ❡ ♣ ✲ ✶ ✿ ❲ ❡ ♠ ❛ ❦ ❡ ❛ ❡ j = 3 ❛ ❜ ♦ ❜ ✐ ♥ ❣ ❧ ✐ ❦ ❡ ✇ ❡ ❞ ✐ ❞ ✐ ♥ ❛ ✭ ❜ ✮ ✳
❙ ❡ ♣ ✲ ✷ ✿ ❲ ❡ ❝ ♦ ♥ ✉ ❝ ❤ ❡ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ① Q ❧ ✐ ❦ ❡ ✇ ❡ ❞ ✐ ❞ ✐ ♥ ❛ ✭ ❜ ✮ ✳
❙ ❡ ♣ ✲ ✸ ✿ ❉ ❡ ✜ ♥ ❡ ❤ ❡ ❛ ❞ ❞ ✐ ✐ ✈ ❡ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ❛ ❧ ❛ f ≡ δ km ✇ ❤ ❡ ❡ ❤ ❡ ❞ ✐ ❛ ❝ ❞ ❡ ❧ ❛ ❢ ✉ ♥ ❝ ✐ ♦ ♥ δ km ❢ ♦ ❛ ✈ ❡ ❝ ♦ k =
k1 k2 · · · kn
✐ ❞ ❡ ✜ ♥ ❡ ❞ ❛
δ km =1 ; m = k1 ♦ m = k2 · · · ♦ m = kn
0 ; ♦ ✳ ✇ ✳
▼ ♦ ❡ ♣ ❡ ❝ ✐ ❡ ❧ ② f ∈ R15✐ ❛ ❝ ♦ ❧ ✉ ♠ ♥ ✈ ❡ ❝ ♦ ✇ ❤ ♦ ❡ 7th ❛ ♥ ❞ 8th ♦ ✇ ✭ ❝ ♦ ❡ ♣ ♦ ♥ ❞ ✐ ♥ ❣ ♦ ❤ ❡ 8th ❛ ♥ ❞ ❤ ❡ 9th ❛ ❡ ❡ ♣ ❡ ❝ ✐ ✈ ❡ ❧ ② ✮
✐ ❡ ♦ 1 ❛ ♥ ❞ ❛ ❧ ❧ ♦ ❤ ❡ ❡ ❧ ❡ ♠ ❡ ♥ ✐ ❡ ♦ 0 ✳
❙ ❡ ♣ ✲ ✹ ✿ ❲ ❡ ❝ ♦ ♠ ♣ ✉ ❡ ❤ ❡ ✈ ❡ ❝ ♦
µshock = (I − Q)−1
f
✶
■ ✐ ❥ ✉ ❛ ❡ ♠ ✇ ❡ ❝ ♦ ✐ ♥ ❡ ❞ ✇ ❤ ✐ ❝ ❤ ✐ ♠ ♣ ❧ ✐ ❡ ❤ ❡ ✉ ❡ ♦ ❢ ❛ ♠ ❡ ♠ ❛ ❤ ❡ ♠ ❛ ✐ ❝ ❛ ❧ ♥ ♦ ❛ ✐ ♦ ♥ ✐ ♥ ♠ ✉ ❧ ✐ ♣ ❧ ❡ ❝ ♦ ♥ ❡ ① ❤ ❛ ✈ ✐ ♥ ❣ ❞ ✐ ✛ ❡ ❡ ♥ ♠ ❡ ❛ ♥ ✐ ♥ ❣ ✦ ✦ ✦
✷
7/23/2019 Solution of Stochastic Process and Modelling
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✇ ❤ ✐ ❝ ❤ ❤ ♦ ✇ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❤ ♦ ❝ ❦ ❘ ❛ ❜ ❡ ❡ ❝ ❡ ✐ ✈ ❡ ❜ ❡ ❢ ♦ ❡ ❡ ❛ ❝ ❤ ✐ ♥ ❣ ❤ ❡ ❝ ❤ ❡ ❡ ❡ ✱ ❛ ✐ ♥ ❣ ❢ ♦ ♠ ❛ ♥ ② ❛ ♥ ✐ ❡ ♥ ❛ ❡ ✳
❙ ❡ ♣ ✲ ✺ ✿ ❲ ❡ ❛ ❡ ✐ ♥ ❡ ❡ ❡ ❞ ✐ ♥ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❤ ♦ ❝ ❦ ❣ ✐ ✈ ❡ ♥ ❤ ❛ ❘ ❛ ❤ ❜ ❡ ❛ ❢ ♦ ♠ ❙ ❛ ❡ ✲ ✶ ✻ ✳ ❚ ❤ ✐ ✐ ❣ ✐ ✈ ❡ ♥ ❜ ②
µshock (15)✱ ❤ ❡ 15th ♦ ✇ ♦ ❢ µshock ✳
❚ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❤ ♦ ❝ ❦ ❘ ❛ ❜ ❡ ❡ ❝ ❡ ✐ ✈ ❡ ❜ ❡ ❢ ♦ ❡ ❡ ❛ ❝ ❤ ✐ ♥ ❣ ❤ ❡ ❝ ❤ ❡ ❡ ❡ ✐ 6.2963✳
✭ ❞ ✮ ❚ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ❝ ❛ ♥ ❜ ❡ ❛ ❜ ❛ ❝ ❡ ❞ ❛ ✿ ❙ ❛ ✐ ♥ ❣ ❢ ♦ ♠ ❛ ❡ i ✱ ✇ ❤ ❛ ✐ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❤ ❛ ❛ ❡ j ✐ ✈ ✐ ✐ ❡ ❞ ❜ ❡ ❢ ♦ ❡ ❛ ❡ ❦
✭ ❝ ❛ ♥ ❜ ❡ ❛ ✈ ❡ ❝ ♦ ✮ ❄ ■ ♥ ❤ ✐ ✉ ❡ ✐ ♦ ♥ i = 16 ✱ j = 3 ❛ ♥ ❞ k = 8 9✳ ❚ ♦ ❛ ♥ ✇ ❡ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ✇ ❡ ❛ ❞ ♦ ♣ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ❡ ♣ ✿
❙ ❡ ♣ ✲ ✶ ✿ ❲ ❡ ♠ ❛ ❦ ❡ ❛ ❡ j = 3 ❛ ♥ ❞ k =
8 9
❛ ❜ ♦ ❜ ✐ ♥ ❣ ✳ ❚ ♦ ❞ ♦ ❤ ✐ ✇ ❡ ♠ ❛ ❦ ❡ ❛ ♠ ♦ ❞ ✐ ❢ ❡ ❞ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ①
P
❛ ❢ ♦ ❧ ❧ ♦ ✇ ✿ ✶ ✮ ❙ ❡
P = P ✳ ✷ ✮ ❙ ❡ ❛ ❧ ❧ ❡ ❧ ❡ ♠ ❡ ♥ ♦ ❢ ❤ ❡ 3rd ✱ 8th ❛ ♥ ❞ 9th ♦ ✇ ♦ ❢
P ❛ 0 ✳ ✸ ✮ ❙ ❡ p33 = p88 = p99 = 1 ✳
❙ ❡ ♣ ✲ ✷ ✿ ❲ ❡ ❝ ♦ ♥ ✉ ❝ ❛ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ① Q ✇ ❤ ✐ ❝ ❤ ❝ ♦ ♥ ❛ ✐ ♥ ❤ ❡ ♦ ♥ ❡ ✲ ❡ ♣ ❛ ♥ ✐ ✐ ♦ ♥ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❢ ♦ ♠ ♦ ♥ ❡ ❛ ♥ ✲
✐ ❡ ♥ ❛ ❡ ♦ ❛ ♥ ♦ ❤ ❡ ❛ ♥ ✐ ❡ ♥ ❛ ❡ ✳ Q ✐ ♦ ❜ ❛ ✐ ♥ ❡ ❞ ❢ ♦ ♠
P ❜ ② ❞ ❡ ❧ ❡ ✐ ♥ ❣ ✐ 3rd ✱ 8th ❛ ♥ ❞ 9th ♦ ✇ ❛ ♥ ❞ 3rd ✱ 8th ❛ ♥ ❞ 9th ❝ ♦ ❧ ✉ ♠ ♥ ✳
❙ ❡ ♣ ✲ ✸ ✿ ❲ ❡ ❝ ♦ ♥ ✉ ❝ ❛ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ① R ✇ ❤ ✐ ❝ ❤ ❝ ♦ ♥ ❛ ✐ ♥ ❤ ❡ ♦ ♥ ❡ ✲ ❡ ♣ ❛ ♥ ✐ ✐ ♦ ♥ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❢ ♦ ♠ ❛ ❛ ♥ ✲
✐ ❡ ♥ ❛ ❡ ♦ ❛ ❡ ❝ ✉ ❡ ♥ ❛ ❡ ✳ Q ✐ ♦ ❜ ❛ ✐ ♥ ❡ ❞ ❢ ♦ ♠
P ❜ ② ❞ ❡ ❧ ❡ ✐ ♥ ❣ ✐ 3rd ✱ 8th ❛ ♥ ❞ 9th ♦ ✇ ❛ ♥ ❞ ❞ ❡ ❧ ❡ ✐ ♥ ❣ ❛ ❧ ❧ ❝ ♦ ❧ ✉ ♠ ♥ ❡ ① ❝ ❡ ♣
❤ ❡ 3rd ✱ 8th ❛ ♥ ❞ 9th ✳
❙ ❡ ♣ ✲ ✹ ✿ ❲ ❡ ❝ ♦ ♠ ♣ ✉ ❡ ❤ ❡ ♠ ❛ ✐ ①
U = (I − Q)−1
R
✇ ❤ ♦ ❡ U ij ❡ ❧ ❡ ♠ ❡ ♥ ✐ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❤ ❛ ❛ ✐ ♥ ❣ ❢ ♦ ♠ ❛ ❛ ♥ ✐ ❡ ♥ ❛ ❡ i✱ ❤ ❡ ♠ ❛ ❦ ♦ ✈ ❝ ❤ ❛ ✐ ♥ ✇ ✐ ❧ ❧ ❣ ❡ ❛ ❜ ♦ ❜ ❡ ❞ ✐ ♥ ❡ ❝ ✉ ❡ ♥
❛ ❡ j ✳
❙ ❡ ♣ ✲ ✺ ✿ ❚ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❤ ❛ ❘ ❛ ❤ ❜ ❡ ❡ ❛ ❝ ❤ ❡ ❝ ❤ ❡ ❡ ❡ ✇ ✐ ❤ ♦ ✉ ❣ ❡ ✐ ♥ ❣ ❤ ♦ ❝ ❦ ❡ ❞ ✐ ❣ ✐ ✈ ❡ ♥ ❜ ② ❤ ❡ (13, 1)th
❡ ❧ ❡ ♠ ❡ ♥ ♦ ❢ U ✳ ■ ♥
❤ ❡ U ♠ ❛ ✐ ① ♦ ✇ 13 ❝ ♦ ❡ ♣ ♦ ♥ ❞ ♦ ❙ ❛ ❡ ✲ ✶ ✻ ❛ ♥ ❞ ❝ ♦ ❧ ✉ ♠ ♥ 1 ❝ ♦ ❡ ♣ ♦ ♥ ❞ ♦ ❙ ❛ ❡ ✲ ✸ ✳
❚ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❤ ❛ ❘ ❛ ❜ ❡ ✜ ♥ ❞ ❤ ❡ ❝ ❤ ❡ ❡ ❡ ❜ ❡ ❢ ♦ ❡ ❣ ❡ ✐ ♥ ❣ ❤ ♦ ❝ ❦ ❡ ❞ ✐ ❢ ♦ ✉ ♥ ❞ ♦ ❜ ❡ 0.2143 ✳
✭ ❡ ✮ ❲ ❡ ✇ ✐ ❧ ❧ ❞ ✐ ✈ ✐ ❞ ❡ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ✐ ♥ ✇ ♦ ♣ ❛ ✿ ✐ ✮ ❋ ♦ ♠ ✉ ❧ ❛ ✐ ♥ ❣ ❤ ❡ ▼ ❛ ❦ ♦ ✈ ❈ ❤ ❛ ✐ ♥ ▼ ♦ ❞ ❡ ❧ ♦ ❢ ❤ ❡ ❘ ❛ ❜ ❡ ✲ ❙ ❛ ♠ ② ❡ ♠ ✳ ✐ ✐ ✮
❈ ❛ ❧ ❝ ✉ ❧ ❛ ✐ ♥ ❣ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ✐ ♠ ❡ ♦ ✇ ❛ ✐ ✉ ♥ ✐ ❧ ❙ ❛ ♠ ❛ ♥ ❞ ❘ ❛ ❜ ❡ ✜ ♥ ❞ ❡ ❛ ❝ ❤ ♦ ❤ ❡ ✳
❋ ♦ ♠ ✉ ❧ ❛ ✐ ♥ ❣ ❤ ❡ ▼ ❛ ❦ ♦ ✈ ❈ ❤ ❛ ✐ ♥ ▼ ♦ ❞ ❡ ❧ ♦ ❢ ❤ ❡ ❘ ❛ ❜ ❡ ✲ ❙ ❛ ♠ ② ❡ ♠ ✿
❚ ❤ ✐ ❡ ♣ ❝ ❛ ♥ ❜ ❡ ❢ ✉ ❤ ❡ ✉ ❜ ✲ ❞ ✐ ✈ ✐ ❞ ❡ ❞ ✐ ♥ ♦ ✇ ♦ ❡ ♣ ✿ ■ ✮ ▼ ♦ ❞ ❡ ❧ ❧ ✐ ♥ ❣ ❤ ❡ ♠ ♦ ✈ ❡ ♠ ❡ ♥ ♦ ❢ ❙ ❛ ♠ ❛ ❧ ♦ ♥ ❡ ✉ ✐ ♥ ❣ ❛ ▼ ❛ ❦ ♦ ✈ ❈ ❤ ❛ ✐ ♥
♠ ♦ ❞ ❡ ❧ ✳ ■ ■ ✮ ▼ ♦ ❞ ❡ ❧ ❧ ✐ ♥ ❣ ❤ ❡ ❝ ♦ ♠ ❜ ✐ ♥ ❡ ❞ ♠ ♦ ✈ ❡ ♠ ❡ ♥ ♦ ❢ ❘ ❛ ❜ ❡ ✲ ❙ ❛ ♠ ② ❡ ♠ ✉ ✐ ♥ ❣ ▼ ❛ ❦ ♦ ✈ ❈ ❤ ❛ ✐ ♥ ♠ ♦ ❞ ❡ ❧ ✳
❲ ❡ ✇ ✐ ❧ ❧ ✜ ♠ ♦ ❞ ❡ ❧ ❤ ❡ ♠ ♦ ✈ ❡ ♠ ❡ ♥ ♦ ❢ ❙ ❛ ♠ ❛ ❧ ♦ ♥ ❡ ✳ ▼ ♦ ✈ ❡ ♠ ❡ ♥ ♦ ❢ ❙ ❛ ♠ ✐ ♥ ❤ ❡ ♠ ❛ ③ ❡ ✐ ❣ ✉ ✐ ❞ ❡ ❞ ❜ ② ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ❤ ❡ ❡ ✉ ❧ ❡ ✿
• ❘ ✉ ❧ ❡ ✲ ✶ ✿ ■ ♥ ❡ ✈ ❡ ② ❡ ♣ ♦ ❝ ❤ ❤ ❡ ❡ ✐ ❛ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ♦ ❢ 0.2 ❤ ❛ ❙ ❛ ♠ ✇ ♦ ♥ ✬ ♠ ♦ ✈ ❡ ❜ ❡ ❝ ❛ ✉ ❡ ❤ ❡ ✴ ❤ ❡ ✐ ✐ ❡ ❞ ❛ ♥ ❞ ✇ ❛ ♥ ♦ ❡ ✳
• ❘ ✉ ❧ ❡ ✲ ✷ ✿ ■ ♥ ❡ ✈ ❡ ② ❡ ♣ ♦ ❝ ❤ ✱ ❙ ❛ ♠ ❝ ❛ ♥ ♠ ♦ ✈ ❡ ♦ ♥ ❧ ② ♦ ♥ ❡ ❡ ♣ ❡ ✐ ❤ ❡ ✉ ♣ ♦ ❞ ♦ ✇ ♥ ♦ ❧ ❡ ❢ ♦ ✐ ❣ ❤ ✳ ❉ ✐ ❛ ❣ ♦ ♥ ❛ ❧ ♠ ♦ ✈ ❡ ♠ ❡ ♥ ✐ ◆ ❖ ❚
❛ ❧ ❧ ♦ ✇ ❡ ❞ ✳
• ❘ ✉ ❧ ❡ ✲ ✸ ✿ ❙ ✐ ♥ ❝ ❡ ❙ ❛ ♠ ✐ ❜ ❧ ✐ ♥ ❞ ✱ ✐ ❝ ❛ ♥ ② ♦ ♠ ♦ ✈ ❡ ✐ ♥ ❛ ♥ ② ♦ ♥ ❡ ♦ ❢ ❤ ❡ ❢ ♦ ✉ ❞ ✐ ❡ ❝ ✐ ♦ ♥ ✇ ✐ ❤ ❡ ✉ ❛ ❧ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ✳ ■ ❢ ❛ ❛ ❣ ✐ ✈ ❡ ♥
❡ ♣ ♦ ❝ ❤ ❙ ❛ ♠ ✐ ❡ ♦ ♠ ♦ ✈ ❡ ✐ ♥ ❤ ❡ ❞ ✐ ❡ ❝ ✐ ♦ ♥ ✇ ❤ ✐ ❝ ❤ ❤ ❛ ❛ ♥ ♦ ❜ ❛ ❝ ❧ ❡ ✱ ❤ ❡ ♥ ✐ ✇ ✐ ❧ ❧ ❣ ❡ ❜ ♦ ✉ ♥ ❝ ❡ ❞ ❜ ❛ ❝ ❦ ❛ ♥ ❞ ❤ ❡ ♥ ❝ ❡ ✇ ✐ ❧ ❧ ❡ ♠ ❛ ✐ ♥
✐ ♥ ❤ ❡ ❛ ♠ ❡ ❛ ❡ ✳
❙ ❛ ② ❤ ❛ ✐ ♥ ❛ ❣ ✐ ✈ ❡ ♥ ❛ ❡ ✱ ❙ ❛ ♠ ❝ ❛ ♥ ♠ ♦ ✈ ❡ ✐ ♥ m ∈ {1, 2, 3, 4} ♣ ♦ ✐ ❜ ❧ ❡ ❞ ✐ ❡ ❝ ✐ ♦ ♥ ✳ ❚ ❤ ❡ ♥ ✉ ♥ ❞ ❡ ❤ ❡ ✉ ❧ ❡ ❞ ❡ ✜ ♥ ❡ ❞ ❛ ❜ ♦ ✈ ❡ ✱ ❤ ❡
♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❤ ❛ ❙ ❛ ♠ ❡ ♠ ❛ ✐ ♥ ✐ ♥ ❤ ❡ ❛ ♠ ❡ ❛ ❡ ✐ 0.2 + (1−0.2)4 (4 − m) = 1 − 0.2m ❛ ♥ ❞ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❤ ❛ ❙ ❛ ♠ ♠ ♦ ✈ ❡
✐ ♥ ♦ ♥ ❡ ♦ ❢ ❤ ❡ m ♣ ♦ ✐ ❜ ❧ ❡ ❞ ✐ ❡ ❝ ✐ ♦ ♥ ✐
(1−0.2)4 = 0.2 ✳ ❚ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❛ ♥ ✐ ✐ ♦ ♥ ♠ ❛ ✐ ① W ❣ ♦ ✈ ❡ ♥ ✐ ♥ ❣ ❤ ❡ ♠ ♦ ✈ ❡ ♠ ❡ ♥ ♦ ❢ ❙ ❛ ♠
✐ ❤ ♦ ✇ ♥ ❜ ❡ ❧ ♦ ✇ ✳
✸
7/23/2019 Solution of Stochastic Process and Modelling
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✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶ ✵ ✶ ✶ ✶ ✷ ✶ ✸ ✶ ✹ ✶ ✺ ✶ ✻
✶ ✵ ✳ ✽ ✵ ✵ ✵ ✵ ✳ ✷ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵
✷ ✵ ✵ ✳ ✻ ✵ ✳ ✷ ✵ ✵ ✵ ✳ ✷ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵
✸ ✵ ✵ ✳ ✷ ✵ ✳ ✻ ✵ ✳ ✷ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵
✹ ✵ ✵ ✵ ✳ ✷ ✵ ✳ ✻ ✵ ✵ ✵ ✵ ✳ ✷ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵
✺ ✵ ✳ ✷ ✵ ✵ ✵ ✵ ✳ ✻ ✵ ✳ ✷ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵
✻ ✵ ✵ ✳ ✷ ✵ ✵ ✵ ✳ ✷ ✵ ✳ ✷ ✵ ✳ ✷ ✵ ✵ ✵ ✳ ✷ ✵ ✵ ✵ ✵ ✵ ✵
✼ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✷ ✵ ✳ ✻ ✵ ✳ ✷ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵
✽ ✵ ✵ ✵ ✵ ✳ ✷ ✵ ✵ ✵ ✳ ✷ ✵ ✳ ✻ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵
✾ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✽ ✵ ✵ ✵ ✵ ✳ ✷ ✵ ✵ ✵
✶ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✷ ✵ ✵ ✵ ✵ ✳ ✻ ✵ ✳ ✷ ✵ ✵ ✵ ✵ ✵
✶ ✶ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✷ ✵ ✳ ✻ ✵ ✵ ✵ ✵ ✳ ✷ ✵
✶ ✷ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✽ ✵ ✵ ✵ ✵ ✳ ✷
✶ ✸ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✷ ✵ ✵ ✵ ✵ ✳ ✻ ✵ ✳ ✷ ✵ ✵
✶ ✹ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✷ ✵ ✳ ✻ ✵ ✳ ✷ ✵
✶ ✺ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✷ ✵ ✵ ✳ ✷ ✵ ✳ ✹ ✵ ✳ ✷
✶ ✻ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✷ ✵ ✵ ✵ ✳ ✷ ✵ ✳ ✻
❚ ❛ ❜ ❧ ❡ ✸ ✿ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ① W ❣ ♦ ✈ ❡ ♥ ✐ ♥ ❣ ❤ ❡ ♠ ♦ ✈ ❡ ♠ ❡ ♥ ♦ ❢ ❙ ❛ ♠ ✐ ♥ ❤ ❡ ♠ ❛ ③ ❡ ✳
◆ ♦ ✇ ❝ ♦ ♥ ✐ ❞ ❡ ❤ ❡ ❝ ♦ ♠ ❜ ✐ ♥ ❡ ❞ ♠ ♦ ✈ ❡ ♠ ❡ ♥ ♦ ❢ ❘ ❛ ❜ ❡ ❛ ♥ ❞ ❙ ❛ ♠ ✳ ❋ ♦ ❡ ❛ ❝ ❤ ♦ ❢ ❤ ❡ 16 ❛ ❡ ♦ ❢ ❘ ❛ ❜ ❡ ✱ ❙ ❛ ♠ ❝ ❛ ♥ ❜ ❡ ✐ ♥ 16 ♣ ♦ ✐ ❜ ❧ ❡
❛ ❡ ✳ ❍ ❡ ♥ ❝ ❡ ❤ ❡ ❡ ✉ ✐ ❡ ❞ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❛ ❡ ✐ ♥ ❤ ❡ ♠ ❛ ❦ ♦ ✈ ❝ ❤ ❛ ✐ ♥ ♦ ❝ ❛ ♣ ✉ ❡ ❤ ✐ ♠ ♦ ❞ ❡ ❧ ✐
16 × 16 = 256✳ ❲ ❡ ❤ ❛ ✈ ❡ ♦ ✜
❣ ✐ ✈ ❡ ♣ ❤ ② ✐ ❝ ❛ ❧ ♠ ❡ ❛ ♥ ✐ ♥ ❣ ♦ ❤ ❡ ❡ ❛ ❡ ✳ ■ ❢ ❘ ❛ ❜ ❡ ✐ ✐ ♥ ❛ ❡ sr ∈ {1, 2, . . . , 16} ❛ ♥ ❞ ❙ ❛ ♠ ✐ ✐ ♥ ❛ ❡ ss ∈ {1, 2, . . . , 16}✱ ❤ ❡ ♥
✐ ♥ ❤ ❡ ❝ ♦ ♠ ❜ ✐ ♥ ❡ ❞ ♠ ❛ ❦ ♦ ✈ ❝ ❤ ❛ ✐ ♥ ❤ ❡ ❛ ❡ ♦ ❢ ❤ ❡ ❘ ❛ ❜ ❡ ✲ ❙ ❛ ♠ ② ❡ ♠ ✐
i = 16 (sr − 1) + (ss − 1) ✭ ✶ ✮
◆ ♦ ❡ ❤ ❛ i ∈ {0, 1, 2, . . . , 255}✳ ❲ ❡ ✇ ✐ ❧ ❧ ♥ ♦ ✇ ❝ ♦ ♥ ❝ ❡ ♥ ❛ ❡ ♦ ♥ ❞ ❡ ✐ ❣ ♥ ✐ ♥ ❣ ❤ ❡ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ① M ♦ ❢ ❤ ❡
❝ ♦ ♠ ❜ ✐ ♥ ❡ ❞ ♠ ❛ ❦ ♦ ✈ ❝ ❤ ❛ ✐ ♥ ✳ ❆ ✉ ♠ ✐ ♥ ❣ ❤ ❛ ❙ ❛ ♠ ❛ ♥ ❞ ❘ ❛ ❜ ❡ ✬ ♠ ♦ ✈ ❡ ♠ ❡ ♥ ✐ ✐ ♥ ❞ ❡ ♣ ❡ ♥ ❞ ❡ ♥ ♦ ❢ ❡ ❛ ❝ ❤ ♦ ❤ ❡ ✱ ❤ ❡ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ②
❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ① M ✐ ❣ ✐ ✈ ❡ ♥ ❜ ②
M ij = P sirs
jr
W siss
js
✭ ✷ ✮
✇ ❤ ❡ ❡ sir ❛ ♥ ❞ sjr ✭ sis ❛ ♥ ❞ sjs ✮ ✐ ❤ ❡ ❛ ❡ ♦ ❢ ❘ ❛ ❜ ❡ ✭ ❙ ❛ ♠ ✮ ✇ ❤ ❡ ♥ ❤ ❡ ❛ ❡ ♦ ❢ ❤ ❡ ❘ ❛ ❜ ❡ ✲ ❙ ❛ ♠ ② ❡ ♠ ✐ i ❛ ♥ ❞ j ❡ ♣ ❡ ❝ ✐ ✈ ❡ ❧ ② ✳
▼ ❛ ❤ ❡ ♠ ❛ ✐ ❝ ❛ ❧ ❧ ② ✱
sir = ✢ ♦ ♦
i
16
+ 1 ✭ ✸ ✮
sis = i − 16
sir − 1
+ 1 ✭ ✹ ✮
sjr = ✢ ♦ ♦
j
16
+ 1 ✭ ✺ ✮
sjs = j − 16
sjr − 1
+ 1 ✭ ✻ ✮
❊ ✉ ❛ ✐ ♦ ♥ ✷ ❛ ❧ ♦ ♥ ❣ ✇ ✐ ❤ ❡ ✉ ❛ ✐ ♦ ♥ ✸ ✱ ✹ ✱ ✺ ❛ ♥ ❞ ✻ ❝ ♦ ♠ ♣ ❧ ❡ ❡ ❧ ② ❝ ❛ ♣ ✉ ❡ ❤ ❡ ♠ ❛ ❦ ♦ ✈ ❝ ❤ ❛ ✐ ♥ ♠ ♦ ❞ ❡ ❧ ♦ ❢ ❤ ❡ ❘ ❛ ❜ ❡ ✲ ❙ ❛ ♠ ② ❡ ♠ ✳
❈ ❛ ❧ ❝ ✉ ❧ ❛ ✐ ♥ ❣ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ✐ ♠ ❡ ♦ ✇ ❛ ✐ ✉ ♥ ✐ ❧ ❙ ❛ ♠ ❛ ♥ ❞ ❘ ❛ ❜ ❡ ✜ ♥ ❞ ❡ ❛ ❝ ❤ ♦ ❤ ❡ ✿
❋ ✐ ♦ ❢ ❛ ❧ ❧ ❧ ❡ ✜ ♥ ❞ ❤ ❡ ✐ ♥ ✐ ✐ ❛ ❧ ❛ ❡ ♦ ❢ ❤ ❡ ❝ ♦ ♠ ❜ ✐ ♥ ❡ ❞ ♠ ❛ ❦ ♦ ✈ ❝ ❤ ❛ ✐ ♥ ✳ ❲ ❡ ❦ ♥ ♦ ✇ ❤ ❡ ❢ ♦ n = 0 ✱ sr = 16 ❛ ♥ ❞ ss = 1 ✳ ❲ ❤ ❡ ♥
✉ ❜ ✐ ✉ ❡ ❞ ✐ ♥ ❡ ✉ ❛ ✐ ♦ ♥ ✶ ✇ ❡ ✜ ♥ ❞ ❤ ❛ ❤ ❡ ✐ ♥ ✐ ✐ ❛ ❧ ❛ ❡ ♦ ❢ ❤ ❡ ❝ ♦ ♠ ❜ ✐ ♥ ❡ ❞ ▼ ❛ ❦ ♦ ✈ ❈ ❤ ❛ ✐ ♥ ✐ i = 240 ✳ ❖ ✉ ♦ ❢ ❤ ❡ 256 ❛ ❡
❤ ❡ ❡ ❛ ❡ ♦ ♥ ❧ ② 16 ❛ ❡ ✇ ❤ ✐ ❝ ❤ ❝ ♦ ❡ ♣ ♦ ♥ ❞ ♦ ❙ ❛ ♠ ❛ ♥ ❞ ❘ ❛ ❜ ❡ ♠ ❡ ❡ ✐ ♥ ❣ ❡ ❛ ❝ ❤ ♦ ❤ ❡ ✳ ❚ ❤ ❡ ❡ ❛ ❡ ❛ ❡ ❝ ❤ ❛ ❛ ❝ ❡ ✐ ❡ ❞ ❜ ②
sr = ss ✳ ❯ ✐ ♥ ❣ ❡ ✉ ❛ ✐ ♦ ♥ ✶ ✱ ✇ ❡ ❝ ❛ ♥ ❛ ② ❤ ❛ ❙ ❛ ♠ ❛ ♥ ❞ ❘ ❛ ❜ ❡ ♠ ❡ ❡ ❡ ❛ ❝ ❤ ♦ ❤ ❡ ✐ ❢ ❛ ❡ j ♦ ❢ ❤ ❡ ❝ ♦ ♠ ❜ ✐ ♥ ❡ ❞ ♠ ❛ ❦ ♦ ✈ ❝ ❤ ❛ ✐ ♥
❜ ❡ ❧ ♦ ♥ ❣ ♦ ❤ ❡ ❡ C = {0, 17, 34, . . . , 255}✳
■ ♥ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ✇ ❡ ❛ ❡ ✐ ♥ ❡ ❡ ❡ ❞ ✐ ♥ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ✐ ♠ ❡ ♦ ❡ ❛ ❝ ❤ ❛ ♥ ② ♦ ♥ ❡ ♦ ❢ ❤ ❡ ❛ ❡ ✐ ♥ C ✳ ❚ ♦ ❝ ❛ ❧ ❝ ✉ ❧ ❛ ❡ ❤ ✐ ✇ ❡ ❝ ❛ ♥ ♠ ❛ ❦ ❡
❡ ❛ ❝ ❤ ❛ ❡ ✐ ♥ C ❛ ❛ ❝ ❧ ♦ ❡ ❞ ✭ ❡ ❝ ✉ ❡ ♥ ✮ ❝ ♦ ♠ ♠ ✉ ♥ ✐ ❝ ❛ ✐ ♦ ♥ ❝ ❧ ❛ ❛ ♥ ❞ ❤ ❡ ♥ ❝ ❛ ❧ ❝ ✉ ❧ ❛ ❡ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ✐ ♠ ❡ ♣ ❡ ♥ ✐ ♥ ❤ ❡ ❛ ♥ ✐ ❡ ♥
❛ ❡ ✳ ❲ ❡ ❝ ❛ ♥ ✉ ♠ ♠ ❛ ✐ ③ ❡ ❤ ❡ ❡ ✉ ✐ ❡ ❞ ❡ ♣ ❛ ❢ ♦ ❧ ❧ ♦ ✇ ✿
❙ ❡ ♣ ✲ ✶ ✿ ▼ ❛ ❦ ❡ ❛ ♠ ♦ ❞ ✐ ✜ ❡ ❞ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ①
M ✉ ❝ ❤ ❤ ❛ ❛ ❧ ❧ ❤ ❡ ❛ ❡ ✐ ♥ ❡ C ❛ ❡ ❛ ❜ ♦ ❜ ✐ ♥ ❣ ✳
✹
7/23/2019 Solution of Stochastic Process and Modelling
http://slidepdf.com/reader/full/solution-of-stochastic-process-and-modelling 5/31
❙ ❡ ♣ ✲ ✷ ✿ ❡ ❢ ♦ ♠ ❤ ❡ ❝ ❛ ♥ ♦ ♥ ✐ ❝ ❛ ❧ ❞ ❡ ❝ ♦ ♠ ♣ ♦ ✐ ✐ ♦ ♥ ♦ ❢
M ♦ ❣ ❡ ♥ ❡ ❛ ❡ ❤ ❡ Q ♠ ❛ ✐ ① ✳ ● ✐ ✈ ❡ ♥ ❤ ❛ ❤ ❡ ❡ ❛ ❡ 16 ❛ ❜ ♦ ❜ ✐ ♥ ❣ ❛ ❡ ✱
❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❛ ♥ ✐ ❡ ♥ ❛ ❡ ✐ 256 − 16 = 240 ✳ ❍ ❡ ♥ ❝ ❡ Q ∈ R240×240
✳
❙ ❡ ♣ ✲ ✸ ✿ ❈ ❛ ❧ ❝ ✉ ❧ ❛ ❡ µτ = (I − Q)−1
f ✇ ❤ ❡ ❡ f ∈ R240×1✐ ❛ ❝ ♦ ❧ ✉ ♠ ♥ ✈ ❡ ❝ ♦ ✇ ✐ ❤ ❛ ❧ ❧ ❡ ♥ ✐ ❡ ❛ 1 ✳
❙ ❡ ♣ ✲ ✹ ✿ ■ ♥ ❤ ❡ ❝ ♦ ❧ ✉ ♠ ♥ ✈ ❡ ❝ ♦ µτ ✱ ❤ ❡ ✐ ♥ ✐ ✐ ❛ ❧ ❛ ❡ ♦ ❢ ❤ ❡ ② ❡ ♠ i = 240 ✱ ✐ ❛ ♦ ❝ ✐ ❛ ❡ ❞ ✇ ✐ ❤ ♦ ✇ ♥ ✉ ♠ ❜ ❡ 226✳ ❍ ❡ ♥ ❝ ❡ ❤ ❡
❡ ✉ ✐ ❡ ❞ ❛ ♥ ✇ ❡ ✐ µτ (226)✳
❚ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ✐ ♠ ❡ ♦ ✇ ❛ ✐ ✉ ♥ ✐ ❧ ❙ ❛ ♠ ❛ ♥ ❞ ❘ ❛ ❜ ❡ ✜ ♥ ❞ ❡ ❛ ❝ ❤ ♦ ❤ ❡ ✐ ❡ ✉ ❛ ❧ ♦ 54.4488 ✳
✭ ❢ ✮ ■ ♥ ❛ ✭ ❜ ✮ ✱ ✇ ❡ ❢ ♦ ✉ ♥ ❞ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ✐ ♠ ❡ ❢ ♦ ❘ ❛ ❜ ❡ ♦ ✜ ♥ ❞ ❤ ❡ ❝ ❤ ❡ ❡ ❡ ❣ ✐ ✈ ❡ ♥ ❤ ❛ ✐ ❛ ❛ ❙ ❛ ❡ ✶ ✻ ✳ ❚ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞
✐ ♠ ❡ ✐ 85.1852✳ ❆ ❢ ❡ ❡ ❛ ✐ ♥ ❣ ❤ ❡ ❝ ❤ ❡ ❡ ❡ ❢ ♦ ❤ ❡ 1st ✐ ♠ ❡ ✱ ❘ ❛ ❜ ❡ ❤ ❛ ♦ ❡ ❛ ❝ ❤ ❡ ❡ ❡ C − 1 ✐ ♠ ❡ ❜ ❡ ❢ ♦ ❡ ✐ ❣ ♦ ❡ ♦ ❡ ✳ ❚ ❤ ❡
❡ ① ♣ ❡ ❝ ❡ ❞ ✐ ♠ ❡ ♦ ✜ ♥ ❞ ❤ ❡ ❝ ❤ ❡ ❡ ❡ ❢ ♦ ❤ ❡ ❧ ❛ C − 1 ✐ ♠ ❡ ❛ ❡ ❡ ✉ ❛ ❧ ✳ ▲ ❡ ❤ ✐ ✐ ♠ ❡ ❜ ❡ τ ✳ ❖ ♥ ❡ ✐ ❝ ❦ ♦ ✜ ♥ ❞ τ ✐ ♦ ♦ ❜ ❡ ✈ ❡
❤ ❛ ✐ ✐ ❤ ❡ ♠ ❡ ❛ ♥ ✜ ❡ ✉ ♥ ✐ ♠ ❡ ♦ ❢ ❛ ❡ ✲ ✸ ✳ ● ✐ ✈ ❡ ♥ ❤ ❛ ❤ ✐ ♠ ❛ ❦ ♦ ✈ ❝ ❤ ❛ ✐ ♥ ✐ ♥ ✐ ❡ ❞ ✉ ❝ ✐ ❜ ❧ ❡ ✱ ✐ ❤ ❛ ❛ ✜ ♥ ✐ ❡ ♠ ❡ ❛ ♥ ❡ ✉ ♥
✐ ♠ ❡ ✳ τ ✐ ✐ ♥ ❞ ❡ ❡ ❞ ❣ ✐ ✈ ❡ ♥ ❜ ② ❛ ✈ ❡ ② ✐ ♠ ♣ ❧ ❡ ❢ ♦ ♠ ✉ ❧ ❛
τ = 1
π3
✇ ❤ ❡ ❡ π3 ✐ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❝ ♦ ❡ ♣ ♦ ♥ ❞ ✐ ♥ ❣ ♦ ❛ ❡ ✸ ♦ ❢ ❤ ❡ ❛ ♦ ❝ ✐ ❛ ❡ ❞ ❛ ✐ ♦ ♥ ❛ ② ❞ ✐ ✐ ❜ ✉ ✐ ♦ ♥ π ✳ ❲ ❡ ❝ ❛ ♥ ✜ ♥ ❞ π ✉ ✐ ♥ ❣ ❤ ❡
❢ ♦ ♠ ✉ ❧ ❛
π = 1T (I − P + ONES )
−1
❯ ✐ ♥ ❣ ▼ ❆ ❚ ▲ ❆ ❇ ✇ ❡ ❢ ♦ ✉ ♥ ❞ ❤ ❛ τ = 16 ✳ ❍ ❡ ♥ ❝ ❡ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ✐ ♠ ❡ ❜ ❡ ❢ ♦ ❡ ❘ ❛ ❜ ❡ ❝ ❛ ♥ ❛ ❦ ❡ ❡ ✐ 85.1852 + 16 (C − 1) ✳
✭ ❣ ✮ ❚ ❤ ❡ ❡ ♣ ♦ ♦ ❧ ✈ ❡ ❤ ✐ ♣ ♦ ❜ ❧ ❡ ♠ ❛ ❡ ❛ ❢ ♦ ❧ ❧ ♦ ✇ ✿
❙ ❡ ♣ ✲ ✶ ✿ ▼ ❛ ❦ ❡ ❛ ♠ ♦ ❞ ✐ ✜ ❡ ❞ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ①
P ✐ ♥ ✇ ❤ ✐ ❝ ❤ ❛ ❡ ✸ ✭ ❝ ❤ ❡ ❡ ❡ ❛ ❡ ✮ ✱ ✽ ❛ ♥ ❞ ✾ ✭ ❤ ♦ ❝ ❦ ❛ ❡ ✮ ❛ ❡
❛ ❜ ♦ ❜ ✐ ♥ ❣ ❛ ❡ ✳
❙ ❡ ♣ ✲ ✷ ✿ ❈ ❛ ❧ ❝ ✉ ❧ ❛ ❡ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ♦ ❢ ❣ ❡ ✐ ♥ ❣ ❛ ❜ ♦ ❜ ❡ ❞ ✐ ♥ ❛ ❡ ✸ ✱ ✽ ♦ ✾ ❛ ✐ ♥ ❣ ❢ ♦ ♠ ❛ ❡ ✶ ✻ ✳ ❚ ❤ ✐ ❝ ❛ ♥ ❜ ❡ ❞ ♦ ♥ ❡ ❜ ② ✉ ✐ ♥ ❣
❤ ❡ ❢ ♦ ♠ ✉ ❧ ❛ U = (I − Q)−1
R ✇ ❤ ❡ ❡ Q ❛ ♥ ❞ R ✐ ♦ ❜ ❛ ✐ ♥ ❡ ❞ ❜ ② ❝ ❛ ♥ ♦ ♥ ✐ ❝ ❛ ❧ ❞ ❡ ❝ ♦ ♠ ♣ ♦ ✐ ✐ ♦ ♥ ♦ ❢
P ✳ ▲ ❡ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ♦ ❢ ❣ ❡ ✐ ♥ ❣
❛ ❜ ♦ ❜ ❡ ❞ ✐ ♥ ❛ ❡ ✸ ✱ ✽ ❛ ♥ ❞ ✾ ❜ ❡ p3 ✱ p8 ❛ ♥ ❞ p9 ✳ ❖ ❜ ❡ ✈ ❡ ❤ ❛ p3 + p8 + p9 = 1 ✳
❙ ❡ ♣ ✲ ✸ ✿ ■ ❢ ❤ ❡ ❛ ❡ ♦ ❢ ❤ ❡ ♠ ♦ ❞ ✐ ✜ ❡ ❞ ♠ ❛ ❦ ♦ ✈ ❝ ❤ ❛ ✐ ♥ ❣ ❡ ❛ ❜ ♦ ❜ ❡ ❞ ✐ ♥ ❛ ❡ ✸ ❤ ❡ ♥ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❤ ♦ ❝ ❦ ✐ 0 ✳ ❚ ❤ ✐ ❤ ❛ ♣ ♣ ❡ ♥
✇ ✐ ❤ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② p3 ✳ ❍ ❡ ♥ ❝ ❡ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❤ ♦ ❝ ❦ ❝ ♦ ❡ ♣ ♦ ♥ ❞ ✐ ♥ ❣ ♦ ❤ ✐ ❝ ❛ ❡ ✐ p3 · 0✳
❙ ❡ ♣ ✲ ✹ ✿ ❙ ❛ ② ❤ ❛ ❤ ❡ ❛ ❡ ♦ ❢ ❤ ❡ ♠ ♦ ❞ ✐ ✜ ❡ ❞ ♠ ❛ ❦ ♦ ✈ ❝ ❤ ❛ ✐ ♥ ❣ ❡ ❛ ❜ ♦ ❜ ❡ ❞ ✐ ♥ ❛ ❡ ✽ ✳ ❚ ❤ ✐ ❤ ❛ ♣ ♣ ❡ ♥ ✇ ✐ ❤ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② p8 ✳ ❚ ♦
❤ ✐ ❡ ♥ ❞ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❤ ♦ ❝ ❦ ❡ ❝ ❡ ✐ ✈ ❡ ❞ ✐ 1 ❛ ♥ ❞ ❤ ❡ ❤ ♦ ❝ ❦ ❣ ❡ ♥ ❡ ❛ ♦ ✐ ♥ ❛ ❡ ✽ ✇ ✐ ❧ ❧ ❜ ❡ ✉ ♥ ❡ ❞ ♦ ✛ ✳ ◆ ♦ ✇ ✇ ❡ ❛ ❦ ❡ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣
✉ ❜ ❡ ♣ ✿
• ❲ ❡ ♠ ❛ ❦ ❡ ❛ ❡ ✽ ❛ ♥ ❞ 9 ✉ ♥ ❛ ❜ ♦ ❜ ✐ ♥ ❣ ✳ ❚ ❤ ✐ ✐ ❞ ♦ ♥ ❡ ❜ ② ❡ ✐ ♥ ❣ ❤ ❡ 8th ❛ ♥ ❞ ❤ ❡ 9th ♦ ✇ ♦ ❢
P ❡ ✉ ❛ ❧ ♦ ❤ ❡ 8th ❛ ♥ ❞ ❤ ❡
9th ♦ ✇ ♦ ❢ P ✳ ❲ ❡ ✐ ❧ ❧ ❦ ❡ ❡ ♣ ❛ ❡ ✸ ❛ ❜ ♦ ❜ ✐ ♥ ❣ ✳
• ◆ ♦ ✇ ✇ ❡ ❤ ❛ ✈ ❡ ♦ ✜ ♥ ❞ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❤ ♦ ❝ ❦ ❡ ❝ ❡ ✐ ✈ ❡ ❞ ✐ ♥ ❛ ❡ ✾ ❜ ❡ ❢ ♦ ❡ ❡ ❛ ❝ ❤ ✐ ♥ ❣ ❤ ❡ ❝ ❤ ❡ ❡ ❡ ✳ ❚ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ❝ ❛ ♥ ❜ ❡
❛ ❜ ❛ ❝ ❡ ❞ ❛ ✿ ❙ ❛ ✐ ♥ ❣ ❢ ♦ ♠ ❛ ❡ ✽ ✱ ✇ ❤ ❛ ❡ ① ♣ ❡ ❝ ❡ ❞ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❡ ♣ ♦ ❝ ❤ ♣ ❡ ♥ ✐ ♥ ❛ ❡ ✾ ✱ ❜ ❡ ❢ ♦ ❡ ❛ ❡ ✸ ✐ ✈ ✐ ✐ ❡ ❞ ❄ ❚ ♦
❛ ♥ ✇ ❡ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ✇ ❡ ✇ ✐ ❧ ❧ ❛ ❦ ❡ ❡ ♣ ✐ ♠ ✐ ❧ ❛ ♦ ❛ ✭ ❝ ✮ ✳
• ▲ ❡ ❤ ❡ ❛ ♥ ✇ ❡ ♦ ❤ ❡ ❛ ❜ ♦ ✈ ❡ ✉ ❡ ✐ ♦ ♥ ❜ ❡ N 8 ✳ ❆ ❧ ♦ ✐ ❡ ❝ ❡ ✐ ✈ ❡ ❞ 1 ❤ ♦ ❝ ❦ ✐ ♥ ❤ ❡ ❜ ❡ ❣ ✐ ♥ ♥ ✐ ♥ ❣ ✳ ❍ ❡ ♥ ❝ ❡ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ♥ ✉ ♠ ❜ ❡
♦ ❢ ❤ ♦ ❝ ❦ ❝ ♦ ❡ ♣ ♦ ♥ ❞ ✐ ♥ ❣ ♦ ❤ ✐ ❝ ❛ ❡ ✐
p8 (N 8 + 1)✳
❙ ❡ ♣ ✲ ✺ ✿ ❙ ❛ ② ❤ ❛ ❤ ❡ ❛ ❡ ♦ ❢ ❤ ❡ ♠ ♦ ❞ ✐ ✜ ❡ ❞ ♠ ❛ ❦ ♦ ✈ ❝ ❤ ❛ ✐ ♥ ❣ ❡ ❛ ❜ ♦ ❜ ❡ ❞ ✐ ♥ ❛ ❡ ✾ ✳ ❚ ❤ ✐ ❤ ❛ ♣ ♣ ❡ ♥ ✇ ✐ ❤ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② p9 ✳ ◆ ♦ ✇
✇ ❡ ❡ ♣ ❡ ❛ ❤ ❡ ❛ ♠ ❡ ❡ ♣ ❛ ❙ ❡ ♣ ✹ ✳ ❚ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❤ ♦ ❝ ❦ ❝ ♦ ❡ ♣ ♦ ♥ ❞ ✐ ♥ ❣ ♦ ❤ ✐ ❝ ❛ ❡ ✐ p9 (N 9 + 1) ✇ ❤ ❡ ❡ N 9 ✐
❤ ❡ ✏ ❊ ① ♣ ❡ ❝ ❡ ❞ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❡ ♣ ♦ ❝ ❤ ♣ ❡ ♥ ✐ ♥ ❛ ❡ ✽ ✱ ❜ ❡ ❢ ♦ ❡ ❛ ❡ ✸ ✐ ✈ ✐ ✐ ❡ ❞ ✱ ❣ ✐ ✈ ❡ ♥ ❤ ❛ ✇ ❡ ❛ ❢ ♦ ♠ ❛ ❡ ✾ ✑ ✳
❋ ✐ ♥ ❛ ❧ ❧ ② ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❤ ♦ ❝ ❦ ✐ p3 · 0 + p8 (N 8 + 1) + p9 (N 9 + 1) = p8 (N 8 + 1) + p9 (N 9 + 1) ✳ ❲ ❡ ❝ ❛ ❧ ❝ ✉ ❧ ❛ ❡ ❞ p8 ✱
p9 ✱ N 8 ❛ ♥ ❞ N 9 ✉ ✐ ♥ ❣ ▼ ❆ ❚ ▲ ❆ ❇ ❛ ♥ ❞ ✜ ♥ ❛ ❧ ❧ ② ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❤ ♦ ❝ ❦ ✇ ❛ ❝ ❛ ❧ ❝ ✉ ❧ ❛ ❡ ❞ ♦ ❜ ❡ 1.791✳ ❆ ♦ ♥ ❡ ♠ ❛ ② ❡ ① ♣ ❡ ❝ ✱
❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❤ ♦ ❝ ❦ ✐ ♥ ❤ ✐ ❝ ❛ ❡ ❤ ♦ ✉ ❧ ❞ ❜ ❡ ❧ ❡ ❤ ❛ ♥ ❤ ❛ ✐ ♥ ❛ ✭ ❝ ✮ ❜ ❡ ❝ ❛ ✉ ❡ ✐ ♥ ❤ ✐ ❝ ❛ ❡ ♦ ♥ ❡ ❤ ♦ ❝ ❦ ❣ ❡ ♥ ❡ ❛ ♦
❣ ❡ ✇ ✐ ❝ ❤ ❡ ❞ ♦ ✛ ✳ ❚ ❤ ✐ ✐ ♥ ✉ ✐ ✐ ♦ ♥ ✐ ❢ ♦ ✉ ♥ ❞ ♦ ❜ ❡ ❝ ♦ ❡ ❝ ❛ 1.791 < 6.2963✳
❆ ❣ ❛ ✐ ♥ ❛ ❧ ❧ ❤ ❡ ▼ ❆ ❚ ▲ ❆ ❇ ❝ ♦ ❞ ❡ ✐ ✐ ♥ ❝ ❧ ✉ ❞ ❡ ❞ ✐ ♥ ❆ ♣ ♣ ❡ ♥ ❞ ✐ ① ❆ ✳
✺
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✶ ✳ ✷ ■ ❲ ❛ ❛ ❙ ♥ ♦ ✇ ❜ ❛ ❧ ❧ ✐ ♥ ❍ ❡ ❧ ❧
❋ ✐ ❣ ✉ ❡ ✶ ✿ ❋ ✐ ❣ ✉ ❡ ❤ ♦ ✇ ✐ ♥ ❣ ♦ ✉ ❣ ♦ ✐ ♥ ❣ ❛ ♥ ❞ ✐ ♥ ❝ ♦ ♠ ✐ ♥ ❣ ❡ ♣ ♦ ❝ ❤ ✳
▲ ❊ ❆ ❙ ❊ ◆ ❖ ❚ ❊ ✿ ❛ ✭ ❛ ✮ ✱ ✭ ❜ ✮ ✱ ✭ ❝ ✮ ❛ ♥ ❞ ✭ ❞ ✮ ❡ ✉ ✐ ❡ ✉ ♦ ✐ ♠ ♣ ❧ ❡ ♠ ❡ ♥ ❤ ❡ ❝ ♦ ❞ ❡ ❛ ♥ ❞ ❡ ❤ ❡ ❝ ♦ ❞ ❡ ✉ ✐ ♥ ❣ ❛ ♥ ❡ ① ❛ ♠ ♣ ❧ ❡ ✳ ❚ ❤ ❡
▼ ❆ ❚ ▲ ❆ ❇ ❝ ♦ ❞ ❡ ❢ ♦ ❤ ❡ ❡ ♣ ❛ ✐ ✐ ♥ ❝ ❧ ✉ ❞ ❡ ❞ ✐ ♥ ❆ ♣ ♣ ❡ ♥ ❞ ✐ ① ❇ ✳ ❆ ❢ ♦ ❤ ❡ ❡ ① ❛ ♠ ♣ ❧ ❡ ✱ ✇ ❡ ❤ ❛ ✈ ❡ ♠ ❛ ❞ ❡ ❛ ❡ ♣ ❡ ❛ ❡ ❡ ❝ ✐ ♦ ♥ ✐ ♥ ❤ ❡ ❡ ♥ ❞
♦ ❢ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ✇ ❤ ❡ ❡ ✇ ❡ ❝ ✉ ♠ ✉ ❧ ❛ ✐ ✈ ❡ ❧ ② ❤ ❛ ♥ ❞ ❧ ❡ ❛ ❧ ❧ ❤ ❡ ❡ ① ❛ ♠ ♣ ❧ ❡ ✳
✭ ❛ ✮ ❋ ✐ ❣ ✉ ❡ ✶ ❤ ♦ ✇ ♦ ✉ ❣ ♦ ✐ ♥ ❣ ❛ ♥ ❞ ✐ ♥ ❝ ♦ ♠ ✐ ♥ ❣ ❡ ♣ ♦ ❝ ❤ ✳ ❖ ✉ ❣ ♦ ✐ ♥ ❣ ❡ ♣ ♦ ❝ ❤ ✐ ❤ ❡ ❡ ♣ ♦ ❝ ❤ ✐ ♥ ✇ ❤ ✐ ❝ ❤ ■ ❡ ♥ ❞ X n ♠ ❛ ✐ ❧ ✳ ■ ♥ ❝ ♦ ♠ ✐ ♥ ❣ ❡ ♣ ♦ ❝ ❤
✐ ❤ ❡ ❡ ♣ ♦ ❝ ❤ ✐ ♥ ✇ ❤ ✐ ❝ ❤ ■ ❡ ❝ ❡ ✐ ✈ ❡ Z n ♠ ❛ ✐ ❧ ✳ ◆ ♦ ✇ ✇ ❡ ✇ ✐ ❧ ❧ ❞ ❡ ✈ ❡ ❧ ♦ ♣ ❤ ❡ ♦ ❝ ❤ ❛ ✐ ❝ ✉ ♣ ❞ ❛ ❡ ✉ ❧ ❡ ♦ ❢ ❤ ✐ ② ❡ ♠ ✳
Z n+1 =
Xni=1
M I n,i ✭ ✼ ✮
X n+1 =
Z n+1j=1
M On+1 , j ✭ ✽ ✮
■ ♥ ❡ ✉ ❛ ✐ ♦ ♥ ✼ ✱ M I n,i ✐ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ✐ ♥ ❝ ♦ ♠ ✐ ♥ ❣ ♠ ❛ ✐ ❧ ❝ ♦ ❡ ♣ ♦ ♥ ❞ ✐ ♥ ❣ ♦ ith ♠ ❛ ✐ ❧ ♦ ❢ ❤ ❡ nth ♦ ✉ ❣ ♦ ✐ ♥ ❣ ❡ ♣ ♦ ❝ ❤ ✳ M I n,i ∼ pI ❛ ♥ ❞
P I (s) ✐ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❣ ❡ ♥ ❡ ❛ ✐ ♥ ❣ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ♦ ❢ pI ✳ ■ ♥ ❡ ✉ ❛ ✐ ♦ ♥ ✼ ✱ M On+1 , j ✐ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ♦ ✉ ❣ ♦ ✐ ♥ ❣ ♠ ❛ ✐ ❧ ❝ ♦ ❡ ♣ ♦ ♥ ❞ ✐ ♥ ❣
♦ jth ♠ ❛ ✐ ❧ ♦ ❢ ❤ ❡ (n + 1)th
✐ ♥ ❝ ♦ ♠ ✐ ♥ ❣ ❡ ♣ ♦ ❝ ❤ ✳ M On+1 , j ∼ pO ❛ ♥ ❞ P O (s) ✐ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❣ ❡ ♥ ❡ ❛ ✐ ♥ ❣ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ♦ ❢ pO ✳ ▲ ❡
P Xn (s) ❛ ♥ ❞ P Z n (s) ❜ ❡ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❣ ❡ ♥ ❡ ❛ ✐ ♥ ❣ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ♦ ❢ ❤ ❡ ❛ ♥ ❞ ♦ ♠ ✈ ❛ ✐ ❛ ❜ ❧ ❡ X n ❛ ♥ ❞ Z n ❡ ♣ ❡ ❝ ✐ ✈ ❡ ❧ ② ✳ ❋ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣
▲ ❡ ❝ ✉ ❡ ✶ ✺ ✭ 5th ◆ ♦ ✈ ❡ ♠ ❜ ❡ ✮ ✇ ❡ ❝ ❛ ♥ ❞ ✐ ❡ ❝ ❧ ② ✇ ✐ ❡ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ❡ ❧ ❛ ✐ ♦ ♥ ❤ ✐ ♣
P Z n+1 (s) = P Xn (P I (s)) ✭ ✾ ✮
P Xn+1 (s) = P Z n+1 (P O (s)) ✭ ✶ ✵ ✮
❯ ✐ ♥ ❣ ❡ ✉ ❛ ✐ ♦ ♥ ✾ ❛ ♥ ❞ ✶ ✵ ✇ ❡ ❣ ❡
P Xn+1 (s) = P Xn (P I (P O (s))) ✭ ✶ ✶ ✮
▲ ❡ P Y (s) = P I (P O (s)) ✭ ♦ P Y = P I ◦ P O ✮ ✳ ❆ ❧ ♦ ❧ ❡ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ♠ ❛ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ❝ ♦ ❡ ♣ ♦ ♥ ❞ ✐ ♥ ❣ ♦ P Y (s) ❜ ❡ pY ✳ ❚ ❤ ❡ ♥
❤ ❡ ❡ ❧ ❛ ✐ ♦ ♥ ❜ ❡ ✇ ❡ ❡ ♥ X n ❛ ♥ ❞ X n+1 ❝ ❛ ♥ ❜ ❡ ✐ ♠ ♣ ❧ ② ❝ ❛ ♣ ✉ ❡ ❞ ❜ ② ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ♦ ❝ ❤ ❛ ✐ ❝ ✉ ♣ ❞ ❛ ❡ ✉ ❧ ❡
X n+1 =
Xnk=1
Y n,k ✭ ✶ ✷ ✮
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✇ ❤ ❡ ❡ Y n,k ∼ pY ✳ ❚ ❤ ✐ ✐ ❛ ❛ ♥ ❞ ❛ ❞ ● ❛ ❧ ♦ ♥ ✲ ❲ ❛ ♦ ♥ ❇ ❛ ♥ ❝ ❤ ✐ ♥ ❣ ♦ ❝ ❡ ❛ ♥ ❞ ❤ ❡ ♥ ❝ ❡ ✇ ❡ ❝ ❛ ♥ ❛ ♣ ♣ ❧ ② ❤ ❡ ❢ ♦ ♠ ✉ ❧ ❛ ✇ ❤ ✐ ❝ ❤ ✇ ❡
❧ ❡ ❛ ♥ ❡ ❞ ✐ ♥ ▲ ❡ ❝ ✉ ❡ ✶ ✺ ✱ ✶ ✻ ❛ ♥ ❞ ✶ ✼ ✳ ◆ ♦ ✇ ✇ ❡ ♣ ♦ ❝ ❡ ❡ ❞ ✇ ✐ ❤ ❝ ❛ ❧ ❝ ✉ ❧ ❛ ✐ ♥ ❣ ♠ ❡ ❛ ♥ ❛ ♥ ❞ ✈ ❛ ✐ ❛ ♥ ❝ ❡ ♦ ❢ X n ✳
E [X n+1] =
s
d
ds
P Xn+1 (s)
s=1
=
s
d
ds
P Xn
(P I (P O (s)))
s=1
= sP ′
O (s) P ′
I (P O (s)) P ′
Xn (P I (P O (s)))s=1
= P ′
O (1) P ′
I (P O (1)) P ′
Xn (P I (P O (1)))
= P ′
O (1) P ′
I (1) P ′
Xn (P I (1))
= P ′
O (1) P ′
O (1) P ′
Xn (1)
= E [O] E [I ] E [X n]
= µOµI E [X n] ✭ ✶ ✸ ✮
E
(X n+1)
2
=
s
d
ds2
P Xn+1 (s)
s=1
=
s d
ds
sP
′
O (s) P ′
I (P O (s)) P ′
Xn (P I (P O (s)))
s=1
= sP O′
(s) P I ′
P O (s)
P ′
Xn
P I
P O (s)
+s2P ′′
O (s) P ′
I (P O (s)) P ′
Xn (P I (P O (s)))
+s2P ′
O (s) P ′
O (s) P ′′
I (P O (s)) P ′
Xn (P I (P O (s)))
+s2P ′
O (s) P ′
I (P O (s)) P ′
O (s) P ′
I (P O (s)) P ′′
Xn (P I (P O (s)))
s=1
= P ′
O (1) P ′
O (1) P ′
Xn (1) + P
′′
O (1) P ′
I (1) P ′
Xn (1)
+
P ′
O (1)
2
P ′′
I (1) P ′
Xn (1) +
P ′
I (1) P ′
O (1)
2
P ′′
Xn (1)
= P ′O (1) + P ′′O (1)P ′I (1) P ′Xn (1) + P ′O (1)2 P ′′I (1) P ′Xn
(1)
+
P ′
I (1) P ′
O (1)2
P ′′
Xn (1)
= E
O2
E [I ] E [X n] + E [O]2
E
I 2
− E [I ]
E [X n] + (E [I ] E [O])2
E
(X n)2
− E [X n]
=
E
O2
− E [O]2
E [I ] +
E
I 2
− E [I ]2
E [O]2
E [X n] + (E [I ] E [O])2
E
(X n)2
=
❱ ❛ [O] E [I ] + ❱ ❛ [I ] E [O]2
E [X n] + (E [I ] E [O])2
E
(X n)2
=
σ2OµI + σ2
I µ2O
E [X n] + (µOµI )
2E
(X n)2
✭ ✶ ✹ ✮
■ ♥ ❡ ✉ ❛ ✐ ♦ ♥ ✶ ✸ ❛ ♥ ❞ ✶ ✹ ✱ µI ❛ ♥ ❞ σI ✭ µO ❛ ♥ ❞ σO ✮ ✐ ❤ ❡ ♠ ❡ ❛ ♥ ❛ ♥ ❞ ❛ ♥ ❞ ❛ ❞ ❞ ❡ ✈ ✐ ❛ ✐ ♦ ♥ ♦ ❢ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ♠ ❛ ❢ ✉ ♥ ❝ ✐ ♦ ♥ pI ✭ pO ✮
❡ ♣ ❡ ❝ ✐ ✈ ❡ ❧ ② ✳ ❲ ❡ ❝ ❛ ♥ ✇ ✐ ❡ ❡ ✉ ❛ ✐ ♦ ♥ ✶ ✸ ❛ ♥ ❞ ✶ ✹ ✐ ♥ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ♠ ❛ ✐ ① ❢ ♦ ♠ E [X n+1]
E
(X n+1)2
=
µOµI 0
σ2OµI + σ2
I µ2O (µOµI )
2
E [X n]
E
(X n)2
✭ ✶ ✺ ✮
❯ ✐ ♥ ❣ ❡ ✉ ❛ ✐ ♦ ♥ ✶ ✺ ✐ ♥ ❛ ❡ ❝ ✉ ✐ ✈ ❡ ♠ ❛ ♥ ♥ ❡ ✇ ❡ ❣ ❡ E [X n]
E
(X n)2
=
µOµI 0
σ2OµI + σ2
I µ2O (µOµI )
2
n E [X 0]
E
(X 0)2
=
µOµI 0
σ2OµI + σ2
I µ2O (µOµI )
2
n m
m2
✭ ✶ ✻ ✮
❙ ♦ ❧ ✈ ✐ ♥ ❣ ❡ ✉ ❛ ✐ ♦ ♥ ✶ ✻ ✇ ❡ ❣ ❡ ✱
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E [X n] = m (µOµI )n
❱ ❛ [X n] = E
(X n)2
− (E [X n])2
= m
σ2OµI + σ2
I µ2O
(µOµI )
n−1 (µOµI )n
− 1
µOµI − 1
▼ ❆ ❚ ▲ ❆ ❇ ❝ ♦ ❞ ❡ ♦ ❝ ♦ ♠ ♣ ✉ ❡ E [X n] ❛ ♥ ❞ ❱ ❛ [X n] ✐ ✐ ♥ ❝ ❧ ✉ ❞ ❡ ❞ ✐ ♥ ❆ ♣ ♣ ❡ ♥ ❞ ✐ ① ✲ ❇ ✳
✭ ❜ ✮ ❲ ❡ ✇ ✐ ❧ ❧ ❞ ✐ ❡ ❝ ❧ ② ✉ ❡ ❝ ♦ ♥ ❝ ❡ ♣ ❢ ♦ ♠ ▲ ❡ ❝ ✉ ❡ ✶ ✻ ❛ ♥ ❞ ✶ ✼ ♦ ❛ ♥ ✇ ❡ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ✳ ● ✐ ✈ ❡ ♥ ❤ ❛ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ♦ ✉ ❣ ♦ ✐ ♥ ❣
♠ ❛ ✐ ❧ ✐ ♥ ❤ ❡ 0th ❡ ♣ ♦ ❝ ❤ ✐ m ✱ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② pne ❤ ❛ ♦ ✉ ❣ ♦ ✐ ♥ ❣ ♠ ❛ ✐ ❧ ✇ ✐ ❧ ❧ ❜ ❡ ❡ ① ✐ ♥ ❝ ❛ ❤ ❡ nth ❡ ① ❝ ❤ ❛ ♥ ❣ ❡ ✐ ❣ ✐ ✈ ❡ ♥ ❜ ②
pne = am ✇ ❤ ❡ ❡
a = P Y ◦ P Y · · · ◦ P Y (0)
n ❢ ♦ ❧ ❞ ❝ ♦ ♠ ♣ ♦ ✐ ✐ ♦ ♥
❆ ❧ ♦ P Y = P I ◦ P O ✳ ■ ♥ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ♣ ❛ ♦ ❢ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ✇ ❡ ✜ ❧ ❧ ✉ ❡ ❤ ❡ ❤ ♦ ❤ ❛ ♥ ❞ ♥ ♦ ❛ ✐ ♦ ♥ P nY ♦ ❡ ♣ ❡ ❡ ♥ ❤ ❡ n ❢ ♦ ❧ ❞
❝ ♦ ♠ ♣ ♦ ✐ ✐ ♦ ♥ P Y ◦ P Y · · · ◦ P Y ✳
▼ ❆ ❚ ▲ ❆ ❇ ❝ ♦ ❞ ❡ ♦ ❝ ♦ ♠ ♣ ✉ ❡ pne ✐ ✐ ♥ ❝ ❧ ✉ ❞ ❡ ❞ ✐ ♥ ❆ ♣ ♣ ❡ ♥ ❞ ✐ ① ✲ ❇ ✳
✭ ❝ ✮
♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❤ ❛ ■ ❛ ♠ ❢ ♦ ❡ ✈ ❡ ❞ ❡ ❛ ❧ ✐ ♥ ❣ ✇ ✐ ❤ ❡ ♠ ❛ ✐ ❧ ❂ ✶ ✲ ❊ ① ✐ ♥ ❝ ✐ ♦ ♥ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ♦ ❢ ❤ ❡ ❡ ♠ ❛ ✐ ❧
▲ ❡ ❞ ❡ ♥ ♦ ❡ ❡ ① ✐ ♥ ❝ ✐ ♦ ♥ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❜ ② pe ✳ ❯ ✐ ♥ ❣ ❝ ♦ ♥ ❝ ❡ ♣ ❢ ♦ ♠ ▲ ❡ ❝ ✉ ❡ ✶ ✻ ❛ ♥ ❞ ✶ ✼ ✇ ❡ ❝ ❛ ♥ ✇ ✐ ❡
pe = αm✇ ❤ ❡ ❡
α = P Y (α)
❲ ❡ ❝ ❛ ♥ ❛ ❧ ♦ ♦ ❜ ❛ ✐ ♥ α ✉ ✐ ♥ ❣ ❤ ❡ ❢ ♦ ♠ ✉ ❧ ❛
α = limn→∞
P nY (0)
❋ ✐ ♥ ❛ ❧ ❧ ② ✱ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❤ ❛ ■ ❛ ♠ ❢ ♦ ❡ ✈ ❡ ❞ ❡ ❛ ❧ ✐ ♥ ❣ ✇ ✐ ❤ ❡ ♠ ❛ ✐ ❧ ✐ 1 − pe ✳ ▼ ❆ ❚ ▲ ❆ ❇ ❝ ♦ ❞ ❡ ✐ ✐ ♥ ❝ ❧ ✉ ❞ ❡ ❞ ✐ ♥ ❆ ♣ ♣ ❡ ♥ ❞ ✐ ① ✲ ❇ ✳
✭ ❞ ✮ ■ ♥ ❤ ✐ ❝ ❛ ❡ ❤ ❡ ♦ ❝ ❤ ❛ ✐ ❝ ✉ ♣ ❞ ❛ ❡ ✉ ❧ ❡ ❣ ✐ ✈ ❡ ♥ ❜ ② ❡ ✉ ❛ ✐ ♦ ♥ ✶ ✷ ✇ ✐ ❧ ❧ ❣ ❡ ♠ ♦ ❞ ✐ ✜ ❡ ❞ ❛
X n+1 =
Xnk=1
Y n,k + N n
✇ ❤ ❡ ❡ N n ∼ pN ✳ N n ✐ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❛ ❞ ❞ ✐ ✐ ♦ ♥ ❛ ❧ ♦ ✉ ❣ ♦ ✐ ♥ ❣ ♠ ❛ ✐ ❧ ✐ ♥ ❤ ❡ nth ♦ ✉ ❣ ♦ ✐ ♥ ❣ ❡ ♣ ♦ ❝ ❤ ✳ N n ∼ pN ❛ ♥ ❞ P N (s) ✐ ❤ ❡
♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❣ ❡ ♥ ❡ ❛ ✐ ♥ ❣ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ♦ ❢ pN ✳ ◆ ♦ ✇
P Xn+1 (s) = E sXn+1= E
Xnk=1
sY n,k · sN n
= E
Xnk=1
sY n,k
E
sN n
✭ ✶ ✼ ✮
= P Xn (P Y (s)) P N (s) ✭ ✶ ✽ ✮
❊ ✉ ❛ ✐ ♦ ♥ ✶ ✼ ✐ ♣ ♦ ✐ ❜ ❧ ❡ ❞ ✉ ❡ ♦ ❤ ❡ ✐ ♥ ❞ ❡ ♣ ❡ ♥ ❞ ❡ ♥ ❝ ❡ ♦ ❢ ❤ ❡ ❛ ❞ ❞ ✐ ✐ ♦ ♥ ❛ ❧ ♠ ❛ ✐ ❧ N n ❛ ♥ ❞ ❤ ❡ ❡ ❣ ✉ ❧ ❛ ♠ ❛ ✐ ❧ Y n,k ✳
❍ ♦ ✇ ✇ ✐ ❧ ❧ ❛ ♥ ✇ ❡ ♦ ❛ ✭ ❛ ✮ ❝ ❤ ❛ ♥ ❣ ❡ ✿
✽
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❋ ✐ ❧ ❡ ♣ ♦ ❝ ❡ ❡ ❞ ✇ ✐ ❤ ❤ ❡ ♠ ❡ ❛ ♥ ♦ ❢ X n ✳
E [X n+1] =
s
d
ds
P Xn+1 (s)
s=1
=
s
d
ds
P Xn
(P Y (s)) P N (s)
s=1
= sP N (s) d
ds
(P Xn (P Y (s))) + sP Xn
(P Y (s)) d
ds
(P N (s))s=1
✭ ✶ ✾ ✮
= sP N (s) P ′
Y (s) P ′
Xn (P Y (s)) + sP Xn
(P Y (s)) P ′
N (s)s=1
= sP N (s) P ′
Y (s) P ′
Xn (P Y (s))
s=1
+ sP Xn (P Y (s)) P
′
N (s)s=1
= P N (1) sP ′
Y (s) P ′
Xn (P Y (s))
s=1
+ 1 · P Xn (P Y (1)) P
′
N (1)
= 1 · sP ′
Y (s) P ′
Xn (P Y (s))
s=1
+ 1 · P Xn (1) · P
′
N (1)
= µOµI E [X n] + 1 · 1 · P ′
N (1) ✭ ✷ ✵ ✮
= µOµI E [X n] + µN ✭ ✷ ✶ ✮
❊ ✉ ❛ ✐ ♦ ♥ ✶ ✾ ✐ ♦ ❜ ❛ ✐ ♥ ❡ ❞ ✉ ✐ ♥ ❣ ❝ ❤ ❛ ✐ ♥ ✉ ❧ ❡ ❛ ♥ ❞ ❡ ✉ ❛ ✐ ♦ ♥ ✷ ✵ ✐ ♦ ❜ ❛ ✐ ♥ ❡ ❞ ❢ ♦ ♠ ❡ ✉ ❛ ✐ ♦ ♥ ✶ ✸ ✳ ◆ ❡ ① ✇ ❡ ❝ ♦ ♠ ♣ ✉ ❡ ❡ ❝ ♦ ♥ ❞ ♠ ♦ ♠ ❡ ♥
♦ ❢ X n ✇ ❤ ✐ ❝ ❤ ✇ ✐ ❧ ❧ ❜ ❡ ✉ ❡ ❢ ✉ ❧ ♦ ✜ ♥ ❞ ✈ ❛ ✐ ❛ ♥ ❝ ❡ ♦ ❢ X n ✳
E
(X n+1)2
=
s
d
ds
2
P Xn+1 (s)
s=1
=
s
d
ds
sP N (s) P
′
Y (s) P ′
Xn (P Y (s)) + sP Xn
(P Y (s)) P ′
N (s)
s=1
=
s
d
ds
P N (s) sP
′
Y (s) P ′
Xn (P Y (s))
s=1
+
s
d
ds
sP Xn
(P Y (s)) P ′
N (s)
s=1
= s
P N (s)
d
ds
sP
′
Y (s) P ′
Xn (P Y (s))
+ P
′
N (s) sP ′
Y (s) P ′
Xn (P Y (s))
s=1
+ s P Xn (P Y (s)) P ′
N (s) + sP ′
Y (s) P ′
Xn (P Y (s)) P ′
N (s) + sP Xn (P Y (s)) P ′′
N (s)s=1
✭ ✷ ✷ ✮
= P N (1)
s
d
ds
sP
′
Y (s) P ′
Xn (P Y (s))
s=1
+ 1 · P ′
N (1) · 1 · P ′
Y (1) P ′
Xn (P Y (1))
1 · P Xn (P Y (1)) P
′
N (1) + 12P ′
Y (1) P ′
Xn (P Y (1)) P
′
N (1) + 12P Xn (P Y (1)) P
′′
N (1)
=
s
d
ds
sP
′
Y (s) P ′
Xn (P Y (s))
s=1
+ E [N ] E [Y ] E [X n]
+E [N ] + E [N ] E [Y ] E [X n] +
E
N 2
− E [N ]
=
s
d
ds
sP
′
Y (s) P ′
Xn (P Y (s))
s=1
+ 2µN µOµI E [X n] +
σ2N + (µN )
2
✭ ✷ ✸ ✮
= σ2OµI + σ2
I µ2OE [X n] + (µOµI )
2E (X n)
2
✭ ✷ ✹ ✮
+2µN µOµI E [X n] +
σ2N + (µN )
2=
σ2OµI + σ2
I µ2O + 2µN µOµI
E [X n] + (µOµI )
2E
(X n)2
+
σ2N + (µN )
2
✭ ✷ ✺ ✮
❊ ✉ ❛ ✐ ♦ ♥ ✷ ✷ ✐ ♦ ❜ ❛ ✐ ♥ ❡ ❞ ✉ ✐ ♥ ❣ ❝ ❤ ❛ ✐ ♥ ✉ ❧ ❡ ✳ ■ ♥ ❡ ✉ ❛ ✐ ♦ ♥ ✷ ✸ ✇ ❡ ✉ ❡ ❞ ❤ ❡ ❢ ❛ ❝ ❤ ❛ E [Y ] = E [O] E [I ] = µOµI ✳ ❊ ✉ ❛ ✐ ♦ ♥ ✷ ✹
✐ ♦ ❜ ❛ ✐ ♥ ❡ ❞ ❢ ♦ ♠ ❡ ✉ ❛ ✐ ♦ ♥ ✶ ✹ ✳ ❊ ① ♣ ❡ ✐ ♥ ❣ ❡ ✉ ❛ ✐ ♦ ♥ ✷ ✶ ❛ ♥ ❞ ✷ ✺ ✐ ♥ ♠ ❛ ✐ ① ❢ ♦ ♠
E [X n+1]
E
(X n+1)2
=
µOµI 0
σ2OµI + σ2
I µ2O + 2µN µOµI (µOµI )
2
E [X n]
E
(X n)2
+
µN
σ2N + (µN )
2
✭ ✷ ✻ ✮
✾
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❊ ✉ ❛ ✐ ♦ ♥ ✷ ✻ ✐ ❜ ❛ ✐ ❝ ❛ ❧ ❧ ② ❛ ❞ ✐ ❝ ❡ ❡ ✐ ♠ ❡ ▲ ❚ ■ ② ❡ ♠ ✳ ❚ ♦ ❣ ❡ E [X n] ❛ ♥ ❞ E
(X n)2
✇ ❡ ❝ ❛ ♥ ❡ ❝ ✉ ✐ ✈ ❡ ❧ ② ♦ ❧ ✈ ❡ ❡ ✉ ❛ ✐ ♦ ♥ ✷ ✻
✐ ♥ ▼ ❆ ❚ ▲ ❆ ❇ ❛ ✐ ♥ ❣ ❢ ♦ ♠ E [X 0] = m ❛ ♥ ❞ E
(X 0)2
= m2✳ ◗ ✉ ❛ ❧ ✐ ❛ ✐ ✈ ❡ ❧ ② ♣ ❡ ❛ ❦ ✐ ♥ ❣ ✱ E [X n] ❢ ♦ ❤ ❡ ❝ ❛ ❡ ✇ ✐ ❤ ❛ ❞ ❞ ✐ ✐ ♦ ♥ ❛ ❧
♦ ✉ ❣ ♦ ✐ ♥ ❣ ♠ ❛ ✐ ❧ ✐ ❞ ❡ ✜ ♥ ✐ ❡ ❧ ② ❧ ❛ ❣ ❡ ❤ ❛ ♥ E [X n] ✇ ❤ ❡ ♥ ❤ ❡ ❡ ✇ ❤ ❡ ❡ ♥ ♦ ❛ ❞ ❞ ✐ ✐ ♦ ♥ ❛ ❧ ♦ ✉ ❣ ♦ ✐ ♥ ❣ ♠ ❛ ✐ ❧ ✳
❍ ♦ ✇ ✇ ✐ ❧ ❧ ❛ ♥ ✇ ❡ ♦ ❛ ✭ ❜ ✮ ❝ ❤ ❛ ♥ ❣ ❡ ✿
❚ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ❞ ❡ ✐ ✈ ❛ ✐ ♦ ♥ ✐ ✐ ♠ ♣ ♦ ❛ ♥ ♦ ❛ ♥ ✇ ❡ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ✳ ❋ ♦ ♠ ❡ ✉ ❛ ✐ ♦ ♥ ✶ ✽ ✇ ❡ ❤ ❛ ✈ ❡
P Xn (s) = P Xn−1 (P Y (s)) P N (s)
P Xn−1 (s) = P Xn−2 (P Y (s)) P N (s)
P Xn−1 (P Y (s)) = P Xn−2 (P Y (P Y (s))) P N (P Y (s))
P Xn (s) = P Xn−2 (P Y (P Y (s))) P N (P Y (s)) P N (s)
P Xn−2 (s) = P Xn−3 (P Y (s)) P N (s)
P Xn−2 (P Y (P Y (s))) = P Xn−3 (P Y (P Y (P Y (s)))) P N (P Y (P Y (s)))
P Xn (s) = P Xn−3 (P Y (P Y (P Y (s)))) P N (P Y (P Y (s))) P N (P Y (s)) P N (s)
✳
✳
✳
✳
✳
✳
✳
✳
✳
P Xn (s
) = P X0 (P
n
Y (s
))
n−1
k=0 P N P
k
Y (s
)P Xn
(s) = (P nY (s))m
n−1k=0
P N
P kY (s)
✭ ✷ ✼ ✮
❧ ❡ ❛ ❡ ◆ ♦ ❡ ✿ ■ ♥ ❡ ✉ ❛ ✐ ♦ ♥ ✷ ✼ ✱ P kY (s) ❡ ♣ ❡ ❡ ♥ ❤ ❡ k ❢ ♦ ❧ ❞ ❝ ♦ ♠ ♣ ♦ ✐ ✐ ♦ ♥ ♦ ❢ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❣ ❡ ♥ ❡ ❛ ✐ ♥ ❣ ❢ ✉ ♥ ❝ ✐ ♦ ♥ P Y (s)✳
◆ ♦ ✇ ✇ ❡ ✇ ✐ ❧ ❧ ♣ ♦ ❝ ❡ ❡ ❞ ✇ ✐ ❤ ❤ ❡ ♠ ❛ ✐ ♥ ✉ ❡ ✐ ♦ ♥ ✳ ❚ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② pne ❤ ❛ ❤ ❡ ♦ ✉ ❣ ♦ ✐ ♥ ❣ ♠ ❛ ✐ ❧ ✇ ✐ ❧ ❧ ❜ ❡ ❡ ① ✐ ♥ ❝ ❛ ❤ ❡ nth
❡ ① ❝ ❤ ❛ ♥ ❣ ❡ ✐ ❡ ✉ ❛ ❧ ♦ P Xn (0) ✳ ❲ ❡ ❝ ❛ ♥ ❝ ❛ ❧ ❝ ✉ ❧ ❛ ❡ ❤ ✐ ❜ ② ✉ ❜ ✐ ✉ ✐ ♥ ❣ s = 0 ✐ ♥ ❡ ✉ ❛ ✐ ♦ ♥ ✷ ✼ ✳ ◗ ✉ ❛ ❧ ✐ ❛ ✐ ✈ ❡ ❧ ② ✱ pne ✇ ✐ ❤ ❛ ❞ ❞ ✐ ✐ ♦ ♥ ❛ ❧
♦ ✉ ❣ ♦ ✐ ♥ ❣ ♠ ❛ ✐ ❧ ✇ ✐ ❧ ❧ ❞ ❡ ❢ ♥ ✐ ❡ ❧ ② ❜ ❡ ❧ ❡ ❤ ❛ ♥ ✭ ♦ ❡ ✉ ❛ ❧ ♦ ✮ ❤ ❡ ❝ ❛ ❡ ✇ ❤ ❡ ♥ ❤ ❡ ❡ ❛ ❡ ♥ ♦ ✇ ❛ ❞ ❞ ✐ ✐ ♦ ♥ ❛ ❧ ♦ ✉ ❣ ♦ ✐ ♥ ❣ ♠ ❛ ✐ ❧ ✳
❚ ❤ ❡ ▼ ❆ ❚ ▲ ❆ ❇ ❝ ♦ ❞ ❡ ✐ ❣ ✐ ✈ ❡ ♥ ✐ ♥ ❆ ♣ ♣ ❡ ♥ ❞ ✐ ① ✲ ❇ ✳
❍ ♦ ✇ ✇ ✐ ❧ ❧ ❛ ♥ ✇ ❡ ♦ ❛ ✭ ❝ ✮ ❝ ❤ ❛ ♥ ❣ ❡ ✿
♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❤ ❛ ■ ❛ ♠ ❢ ♦ ❡ ✈ ❡ ❞ ❡ ❛ ❧ ✐ ♥ ❣ ✇ ✐ ❤ ❡ ♠ ❛ ✐ ❧ ❂ ✶ ✲ ❊ ① ✐ ♥ ❝ ✐ ♦ ♥ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ♦ ❢ ❤ ❡ ❡ ♠ ❛ ✐ ❧
❚ ❤ ❡ ❡ ① ✐ ♥ ❝ ✐ ♦ ♥ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② pe ✐ ❣ ✐ ✈ ❡ ♥ ❜ ② limn→∞
P Xn (0)
pe = limn→∞
P Xn (0)
= limn→∞
(P nY (0))m
n−1k=0
P N
P kY (0)
❚ ♦ ✜ ♥ ❞ pe ✇ ❡ ❥ ✉ ❛ ❦ ❡ ❛ ❧ ❛ ❣ ❡ n ❛ ♥ ❞ ❝ ♦ ♠ ♣ ✉ ❡ P Xn (0) ✉ ✐ ♥ ❣ ❡ ✉ ❛ ✐ ♦ ♥ ✷ ✼ ✳ ❚ ❤ ❡ ▼ ❆ ❚ ▲ ❆ ❇ ❝ ♦ ❞ ❡ ✐ ❣ ✐ ✈ ❡ ♥ ✐ ♥ ❆ ♣ ♣ ❡ ♥ ❞ ✐ ① ✲ ❇ ✳
✭ ❡ ✮ ❚ ♦ ❛ ♥ ✇ ❡ ❤ ✐ ♣ ❛ ✇ ❡ ❣ ♦ ❜ ❛ ❝ ❦ ♦ ❡ ✉ ❛ ✐ ♦ ♥ ✶ ✽
P Xn+1 (s) = P Xn (P Y (s)) P N (s) ✭ ✷ ✽ ✮
P Xn (s) ✐ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❣ ❡ ♥ ❡ ❛ ✐ ♥ ❣ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ♦ ❢ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ♦ ✉ ❣ ♦ ✐ ♥ ❣ ♠ ❛ ✐ ❧ ✐ ♥ ❤ ❡ nth ❡ ♣ ♦ ❝ ❤ ✳ ■ ❢ ❛ ✐ ♦ ♥ ❛ ② ❞ ✐ ✐ ❜ ✉ ✐ ♦ ♥
❡ ① ✐ ❤ ❡ ♥ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ♠ ❛ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ♦ ❢ X n ✇ ✐ ❧ ❧ ❝ ♦ ♥ ✈ ❡ ❣ ❡ ♦ ❤ ❡ ❛ ✐ ♦ ♥ ❛ ② ❞ ✐ ✐ ❜ ✉ ✐ ♦ ♥ ❢ ♦ ❧ ❛ ❣ ❡ n✳ ❙ ✐ ♥ ❝ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ②
♠ ❛ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ❝ ♦ ♥ ✈ ❡ ❣ ❡ ♦ ❤ ❡ ❛ ✐ ♦ ♥ ❛ ② ❞ ✐ ✐ ❜ ✉ ✐ ♦ ♥ ✱ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❣ ❡ ♥ ❡ ❛ ✐ ♥ ❣ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ✇ ✐ ❧ ❧ ❛ ❧ ♦ ❝ ♦ ♥ ✈ ❡ ❣ ❡ ♦ P π (s)✭ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❣ ❡ ♥ ❡ ❛ ✐ ♥ ❣ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ♦ ❢ ❤ ❡ ❛ ✐ ♦ ♥ ❛ ② ❞ ✐ ✐ ❜ ✉ ✐ ♦ ♥ ✮ ✳ ❚ ❤ ✐ ✐ ❜ ❡ ❝ ❛ ✉ ❡ ❛ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❣ ❡ ♥ ❡ ❛ ✐ ♥ ❣ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ❛ ♥ ❞
❛ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ♠ ❛ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ❤ ❛ ❛ ♦ ♥ ❡ ✲ ♦ ✲ ♦ ♥ ❡ ❡ ❧ ❛ ✐ ♦ ♥ ❤ ✐ ♣ ✳ ▼ ❛ ❤ ❡ ♠ ❛ ✐ ❝ ❛ ❧ ❧ ② ✱ P Xn+1 (s) → P π (s) ❛ ♥ ❞ P Xn (s) → P π (s) ❢ ♦
❧ ❛ ❣ ❡ n ✳ ❙ ✉ ❜ ✐ ✉ ✐ ♥ ❣ ❤ ✐ ✐ ❡ ✉ ❛ ✐ ♦ ♥ ✷ ✽ ✇ ❡ ❣ ❡
✶ ✵
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P π (s) = P π (P Y (s)) P N (s)
P π (s) = P π (P I (P O (s))) P N (s) ✭ ✷ ✾ ✮
❊ ✉ ❛ ✐ ♦ ♥ ✷ ✾ ✐ ❤ ❡ ❡ ✉ ✐ ❡ ❞ ❡ ❧ ❛ ✐ ♦ ♥ ❤ ✐ ♣ ✳
✭ ❢ ✮ ■ ♠ ❛ ② ♥ ♦ ❜ ❡ ♣ ♦ ✐ ❜ ❧ ❡ ♦ ❞ ✐ ❡ ❝ ❧ ② ♦ ❧ ✈ ❡ ❡ ✉ ❛ ✐ ♦ ♥ ✷ ✾ ♦ ❣ ❡ P π (s)✳ ❲ ❡ ✐ ♥ ❡ ❛ ❞ ✉ ❡ ❤ ❡ ✐ ❞ ❡ ❛ ❤ ❛ ✐ ❢ P π (s) ❡ ① ✐ ✱ ❤ ❡ ♥ ✇ ❡
❝ ❛ ♥ ❡ ① ♣ ❡ ✐ ❛ ❛ ❧ ✐ ♠ ✐ ♦ ❢ P Xn (s) ✳ ❯ ✐ ♥ ❣ ❡ ✉ ❛ ✐ ♦ ♥ ✷ ✼ ✇ ❡ ❝ ❛ ♥ ❛ ❧ ❡ ❛ ❞ ② ❡ ① ♣ ❡ P Xn
(s) ✐ ♥ ❡ ♠ ♦ ❢ ❡ ♣ ❡ ❛ ❡ ❞ ❝ ♦ ♠ ♣ ♦ ✐ ✐ ♦ ♥
♦ ❢ P N (s) ✱ P O (s) ❛ ♥ ❞ P I (s)✳ ❍ ❡ ♥ ❝ ❡
P π (s) = limn→∞
P Xn (s)
= limn→∞
(P nY (s))m
n−1k=0
P N
P kY (s)
✭ ✸ ✵ ✮
✇ ❤ ❡ ❡ P Y (s) = P I (P O (s))✳
❊ ❳ ❆ ▼ ▲ ❊
Epoch0 5 10 15 20
M e a n
0
1
2
3
4
5 Without Extra Mails
With Extra Mails
Epoch
0 5 10 15 20
V a r i a n c e
0
2
4
6
8
10
12
14 Without Extra Mails
With Extra Mails
Epoch0 5 10 15 20
E x t i n c t i o n P r o b a b i l i t y
0
0.2
0.4
0.6
0.8
1
Without Extra Mails
With Extra Mails
Epoch
0 5 10 15 20
M e a n
×104
0
1
2
3
4
5
6
7
8Without Extra Mails
With Extra Mails
Epoch0 5 10 15 20
V a r i a n c e
×109
0
0.5
1
1.5
2
2.5Without Extra Mails
With Extra Mails
Epoch0 5 10 15 20
E x t i n c t i o n P r o b a b i l i t y
0
0.01
0.02
0.03
0.04
0.05Without Extra Mails
With Extra Mails
µI µO=0.77µN = 0.67
µN = 0.67
µI µO=1.6
µN = 0.67 µN = 0.67
µI µO=1.6
µN = 0.67
µI µO=0.77µN = 0.67
µI µO=0.77
µI µO=1.6
❋ ✐ ❣ ✉ ❡ ✷ ✿ ❋ ✐ ❣ ✉ ❡ ♦ ✉ ❞ ② ❤ ❡ ❡ ✛ ❡ ❝ ♦ ❢ µOµI ❛ ♥ ❞ ❡ ① ❛ ♠ ❛ ✐ ❧ ♦ ♥ ❤ ❡ ♠ ❡ ❛ ♥ ✱ ✈ ❛ ✐ ❛ ♥ ❝ ❡ ❛ ♥ ❞ ❡ ① ✐ ♥ ❝ ✐ ♦ ♥ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ♦ ❢ ❤ ❡
♥ ✉ ♠ ❜ ❡ ♦ ❢ ♦ ✉ ❣ ♦ ✐ ♥ ❣ ♠ ❛ ✐ ❧ ✳
❲ ❡ ❛ ♥ ❞ ♦ ♠ ❧ ② ❣ ❡ ♥ ❡ ❛ ❡ ❞ ✇ ♦ ❡ ♦ ❢ pO ❛ ♥ ❞ pI ✱ ♦ ♥ ❡ ✇ ✐ ❤ µI µO ≈ 0.8 < 1 ❛ ♥ ❞ ❤ ❡ ♦ ❤ ❡ ✇ ✐ ❤ µI µO ≈ 1.5 > 1 ✳ ❲ ❡ ❛ ❧ ♦
❣ ❡ ♥ ❡ ❛ ❡ ❞ ❛ ❝ ♦ ♠ ♠ ♦ ♥ pN ❢ ♦ ❜ ♦ ❤ ❤ ❡ ❡ ❡ ✇ ✐ ❤ µN ≈ 0.7 ✳ ❲ ❡ ♥ ♦ ✇ ❞ ♦ ✇ ♦ ❦ ✐ ♥ ❞ ♦ ❢ ❝ ♦ ♠ ♣ ❛ ❛ ✐ ✈ ❡ ✉ ❞ ② ✿ ■ ✮ ❚ ❤ ❡ ❡ ✛ ❡ ❝ ♦ ❢
µOµI ✳ ■ ■ ✮ ❚ ❤ ❡ ❡ ✛ ❡ ❝ ♦ ❢ ❡ ① ❛ ♠ ❛ ✐ ❧ ✳
❚ ❤ ❡ ❡ ✛ ❡ ❝ ♦ ❢ µOµI ✿ ❲ ❤ ❡ ♥ µI µO < 1 ✱ ❤ ❡ ♠ ❡ ❛ ♥ ❛ ♥ ❞ ✈ ❛ ✐ ❛ ♥ ❝ ❡ ♦ ❢ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ♦ ✉ ❣ ♦ ✐ ♥ ❣ ♠ ❛ ✐ ❧ ❝ ♦ ♥ ✈ ❡ ❣ ❡ ♦ ❛ ✜ ♥ ✐ ❡ ✈ ❛ ❧ ✉ ❡
❛ ✐ ♠ ❡ ✐ ♥ ❝ ❡ ❛ ❡ ✳ ■ ❢ µI µO > 1 ✱ ❜ ♦ ❤ ♠ ❡ ❛ ♥ ❛ ♥ ❞ ✈ ❛ ✐ ❛ ♥ ❝ ❡ ♦ ❢ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ♦ ✉ ❣ ♦ ✐ ♥ ❣ ♠ ❛ ✐ ❧ ✐ ♥ ❝ ❡ ❛ ❡ ❡ ① ♣ ♦ ♥ ❡ ♥ ✐ ❛ ❧ ❧ ② ✇ ✐ ❤
✐ ♠ ❡ ✳ ❚ ❤ ❡ ❡ ① ✐ ♥ ❝ ✐ ♦ ♥ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ✐ ♥ ❝ ❡ ❛ ❡ ✇ ✐ ❤ ❞ ❡ ❝ ❡ ❛ ❡ ✐ ♥ µOµI ✱ ■ ♥ ❞ ❡ ❡ ❞ ✱ ✐ ❢ ❤ ❡ ❡ ❛ ❡ ♥ ♦ ❡ ① ❛ ♠ ❛ ✐ ❧ ❛ ♥ ❞ µOµI < 1 ❤ ❡ ♥
❡ ① ✐ ♥ ❝ ✐ ♦ ♥ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❝ ♦ ♥ ✈ ❡ ❣ ❡ ♦ 1 ✇ ❤ ✐ ❧ ❡ ✐ ❢ µOµI > 1 ❤ ❡ ♥ ❤ ❡ ❡ ① ✐ ♥ ❝ ✐ ♦ ♥ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❝ ♦ ♥ ✈ ❡ ❣ ❡ ♦ ❛ ✜ ♥ ✐ ❡ ✈ ❛ ❧ ✉ ❡ ✳
❚ ❤ ❡ ❡ ✛ ❡ ❝ ♦ ❢ ❡ ① ❛ ♠ ❛ ✐ ❧ ✿ ❲ ✐ ❤ ❡ ① ❛ ♠ ❛ ✐ ❧ ♠ ❡ ❛ ♥ ❛ ♥ ❞ ✈ ❛ ✐ ❛ ♥ ❝ ❡ ♦ ❢ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ♦ ✉ ❣ ♦ ✐ ♥ ❣ ♠ ❛ ✐ ❧ ✐ ♥ ❝ ❡ ❛ ❡ ✳ ❲ ✐ ❤ ♥ ♦
❡ ① ❛ ♠ ❛ ✐ ❧ ✱ ❤ ❡ ♠ ❡ ❛ ♥ ❛ ♥ ❞ ❤ ❡ ✈ ❛ ✐ ❛ ♥ ❝ ❡ ✇ ❡ ❡ ❝ ♦ ♥ ✈ ❡ ❣ ✐ ♥ ❣ ♦ 0 ✐ ❢ µI µO < 1 ✳ ❍ ♦ ✇ ❡ ✈ ❡ ✇ ✐ ❤ ❡ ① ❛ ♠ ❛ ✐ ❧ ✐ ✐ ❝ ♦ ♥ ✈ ❡ ❣ ✐ ♥ ❣ ♦
❛ ✜ ♥ ✐ ❡ ✈ ❛ ❧ ✉ ❡ ✳ ■ ❢ µOµI < 1 ❤ ❡ ♥ ✇ ✐ ❤ ♥ ♦ ❡ ① ❛ ♠ ❛ ✐ ❧ ❤ ❡ ❡ ① ✐ ♥ ❝ ✐ ♦ ♥ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❝ ♦ ♥ ✈ ❡ ❣ ❡ ♦ 1 ❜ ✉ ✇ ✐ ❤ ❡ ① ❛ ♠ ❛ ✐ ❧ ✐
❝ ♦ ♥ ✈ ❡ ❣ ❡ ♦ ❛ ✜ ♥ ✐ ❡ ✈ ❛ ❧ ✉ ❡ ✳ ■ ❢ µOµI > 1 ❛ ♥ ❞ ❤ ❡ ❡ ❛ ❡ ❡ ① ❛ ♠ ❛ ✐ ❧ ✱ ❤ ❡ ♥ ❤ ❡ ❡ ① ✐ ♥ ❝ ✐ ♦ ♥ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❝ ♦ ♥ ✈ ❡ ❣ ❡ ♦ 0 ✳
✶ ✶
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✷ ◆ ✉ ♠ ❡ ✐ ❝ ❛ ❧ ❈ ♦ ♠ ♣ ✉ ❛ ✐ ♦ ♥
✷ ✳ ✶ ❇ ❛ ♥ ❝ ❤ ✐ ♥ ❣ ❖ ✉ ✐ ♥ ❚ ✇ ♦ ❲ ❛ ②
✭ ❛ ✮ ❚ ❤ ❡ ❙ ♦ ❝ ❤ ❛ ✐ ❝ ❯ ♣ ❞ ❛ ❡ ❘ ✉ ❧ ❡ ❢ ♦ ❤ ❡ ✇ ♦ ✇ ❛ ② ❜ ❛ ♥ ❝ ❤ ✐ ♥ ❣ ♣ ♦ ❝ ❡ ✐
X An+1 =
XAn
kA=1
Y (AA)n,kA
+
XBn
kB=1
Y (BA)n,kB
; X Bn+1 =
XAn
kA=1
Y (AB)n,kA
+
XBn
kB=1
Y (BB)n,kB
✭ ✸ ✶ ✮
✇ ❤ ❡ ❡ X An ❛ ♥ ❞ X Bn ❛ ❡ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ❛ ♥ ❞ ❚ ② ♣ ❡ ✲ ❇ ❡ ♣ ❡ ❝ ✐ ✈ ❡ ❧ ② ✐ ♥ ❤ ❡ nth ❡ ♣ ♦ ❝ ❤ ✳ X An ❛ ♥ ❞ X Bn ❛ ❡ ❛ ❧ ♦
❤ ❡ ❛ ❡ ♦ ❢ ❤ ❡ ▼ ❛ ❦ ♦ ✈ ❈ ❤ ❛ ✐ ♥ ✳ Y (AA)n,kA
✭ Y (AB)n,kA
✮ ✐ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ✭ ❚ ② ♣ ❡ ✲ ❇ ✮ ♣ ♦ ❞ ✉ ❝ ❡ ❞ ❛ ❣ ❡ ♥ kA ♦ ❢ ❚ ② ♣ ❡ ✲ ❆
✐ ♥ ❤ ❡ nth ❡ ♣ ♦ ❝ ❤ ✳ Y (AA)n,kA
❛ ♥ ❞ Y (AB)n,kA
❛ ❡ ❣ ♦ ✈ ❡ ♥ ❡ ❞ ❜ ② ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❞ ✐ ✐ ❜ ✉ ✐ ♦ ♥ p(A) ✱ ✐ ✳ ❡ ✳
Y
(AA)n,kA
, Y (AB)n,kA
∼ p(A)
✳ ❙ ✐ ♠ ✐ ❧ ❛ ❧ ②
Y (BA)n,kB
✭ Y (BB)n,kB
✮ ✐ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ✭ ❚ ② ♣ ❡ ✲ ❇ ✮ ♣ ♦ ❞ ✉ ❝ ❡ ❞ ❛ ❣ ❡ ♥ kB ♦ ❢ ❚ ② ♣ ❡ ✲ ❇ ✐ ♥ ❤ ❡ nth ❡ ♣ ♦ ❝ ❤ ✳ Y (BA)n,kB
❛ ♥ ❞
Y (BB)n,kB
❛ ❡ ❣ ♦ ✈ ❡ ♥ ❡ ❞ ❜ ② ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❞ ✐ ✐ ❜ ✉ ✐ ♦ ♥ p(B)✱ ✐ ✳ ❡ ✳
Y
(BA)n,kB
, Y (BB)n,kB
∼ p(B)
✳
❚ ❤ ❡ ▼ ❆ ❚ ▲ ❆ ❇ ❝ ♦ ❞ ❡ ❢ ♦ ❤ ❡ ▼ ♦ ♥ ❡ ❈ ❛ ❧ ♦ ✐ ♠ ✉ ❧ ❛ ✐ ♦ ♥ ♦ ❢ ❚ ✇ ♦ ❲ ❛ ② ❇ ❛ ♥ ❝ ❤ ✐ ♥ ❣ ♦ ❝ ❡ ✐ ❣ ✐ ✈ ❡ ♥ ✐ ♥ ❆ ♣ ♣ ❡ ♥ ❞ ✐ ① ✲ ❈ ✳
▲ ❊ ❆ ❙ ❊ ◆ ❖ ❚ ❊ ✿ ❘ ❛ ❤ ❡ ❤ ❛ ♥ ❛ ♥ ❞ ♦ ♠ ❧ ② ❞ ♦ ✐ ♥ ❣ ▼ ♦ ♥ ❡ ❈ ❛ ❧ ♦ ✐ ♠ ✉ ❧ ❛ ✐ ♦ ♥ ❢ ♦ ✈ ❛ ✐ ♦ ✉ p(A) ❛ ♥ ❞ p(B) ♦ ❣ ❡ ♥ ❡ ❛ ❡ ❞ ✐ ✛ ❡ ❡ ♥
❝ ❡ ♥ ❛ ✐ ♦ ✱ ✇ ❡ ❤ ♦ ✉ ❣ ❤ ❤ ❛ ✐ ✐ ♠ ♦ ❡ ❛ ❡ ❣ ✐ ❝ ♦ ✜ ❞ ❡ ✈ ❡ ❧ ♦ ♣ ❤ ❡ ❤ ❡ ♦ ② ❜ ② ♦ ❧ ✈ ✐ ♥ ❣ ❛ ✭ ❞ ✮ ✱ ✭ ❡ ✮ ❛ ♥ ❞ ✭ ❢ ✮ ❛ ♥ ❞ ❤ ❡ ♥ ✉ ❡
❤ ❡ ❤ ❡ ♦ ② ♦ ❣ ❡ ♥ ❡ ❛ ❡ ❞ ✐ ✛ ❡ ❡ ♥ ❝ ❡ ♥ ❛ ✐ ♦ ❛ ♥ ❞ ❡ ✐ ✉ ✐ ♥ ❣ ▼ ♦ ♥ ❡ ❈ ❛ ❧ ♦ ✐ ♠ ✉ ❧ ❛ ✐ ♦ ♥ ✳
✭ ❞ ✮ ▲ ❡ P XAn X
Bn
❜ ❡ ❤ ❡ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❉ ✐ ✐ ❜ ✉ ✐ ♦ ♥ ❋ ✉ ♥ ❝ ✐ ♦ ♥ ♦ ❢ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ❛ ❣ ❡ ♥ ✭ X An ✮ ❛ ♥ ❞ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❚ ② ♣ ❡ ✲ ❇
❛ ❣ ❡ ♥ ✭ X Bn ❛ ✮ ✐ ♥ ❤ ❡ nth ❡ ♣ ♦ ❝ ❤ ✳ ▲ ❡ ❤ ❡ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ● ❡ ♥ ❡ ❛ ✐ ♥ ❣ ❋ ✉ ♥ ❝ ✐ ♦ ♥
✷
♦ ❢ P XAn X
Bn
❜ ❡ P XAn+1X
Bn+1
(u, v) ✳ ◆ ♦ ✇ ✱
P XAn+1X
Bn+1
(u, v)
= E
uXAn+1vX
Bn+1
= E
XAn
kA=1
uY (AA)n,kA
XBn
kB=1
uY (BA)n,kB
XAn
kA=1
vY (AB)n,kA
XBn
kB=1
vY (BB)n,kB
✭ ✸ ✷ ✮
= E XAn
kA=1 uY (AA)n,kA v
Y (AB)n,kA
XBn
kB=1 uY (BA)n,kB v
Y (BB)n,kB
=∞
xB=0
∞xA=0
E
xAkA=1
uY (AA)n,kA v
Y (AB)n,kA
xBkB=1
uY (BA)n,kB v
Y (BB)n,kB
|X An = xA, X Bn = xB
P
X An = xA, X Bn = xB
✭ ✸ ✸ ✮
=∞
xB=0
∞xA=0
E
xAkA=1
uY (AA)n,kA v
Y (AB)n,kA
xBkB=1
uY (BA)n,kB v
Y (BB)n,kB
P
X An = xA, X Bn = xB
✭ ✸ ✹ ✮
=∞
xB=0
∞xA=0
xAkA=1
E
uY (AA)n,kA v
Y (AB)n,kA
xBkB=1
E
uY (BA)n,kB v
Y (BB)n,kB
P
X An = xA, X Bn = xB
✭ ✸ ✺ ✮
=∞
xB=0
∞
xA=0 xA
kA=1
E uY (AA)kA v
Y (AB)kA
xB
kB=1
E uY (BA)kB v
Y (BB)kB P X A
n = xA, X B
n = xB ✭ ✸ ✻ ✮
=∞
xB=0
∞xA=0
E
uY (AA)
vY (AB)
xA E
uY (BA)
vY (BB)
xBP
X An = xA, X Bn = xB
✭ ✸ ✼ ✮
=∞
xB=0
∞xA=0
P (A) (u, v)
xA P (B) (u, v)
xBP
X An = xA, X Bn = xB
= P XAn X
Bn
P (A) (u, v) , P (B) (u, v)
✷
❙ ✐ ♥ ❝ ❡ ❤ ❡ ❡ ❛ ❡ ✇ ♦ ❛ ♥ ❞ ♦ ♠ ✈ ❛ ✐ ❛ ❜ ❧ ❡
X An ❛ ♥ ❞
X Bn ✱ ❤ ❡ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ● ❡ ♥ ❡ ❛ ✐ ♥ ❣ ❋ ✉ ♥ ❝ ✐ ♦ ♥ ✇ ✐ ❧ ❧ ❜ ❡ ❝ ❤ ❛ ❛ ❝ ❡ ✐ ③ ❡ ❞ ❜ ② ✇ ♦ ♣ ❛ ❛ ♠ ❡
u❛ ♥ ❞
v
✐ ♥ ❡ ❛ ❞ ♦ ❢ ❛ ✐ ♥ ❣ ❧ ❡ ♣ ❛ ❛ ♠ ❡ ❡ s ✭ ❧ ✐ ❦ ❡ ✇ ❡ ❞ ✐ ❞ ✐ ♥ ❝ ❧ ❛ ✮ ✳
✶ ✷
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❊ ✉ ❛ ✐ ♦ ♥ ✸ ✷ ✐ ♦ ❜ ❛ ✐ ♥ ❡ ❞ ❜ ② ✉ ❜ ✐ ✉ ✐ ♥ ❣ X An+1 ❛ ♥ ❞ X Bn+1 ❢ ♦ ♠ ❡ ✉ ❛ ✐ ♦ ♥ ✸ ✶ ✳ ❚ ♦ ❣ ❡ ❡ ✉ ❛ ✐ ♦ ♥ ✸ ✸ ✱ ✇ ❡ ✉ ❡ ❤ ❡ ▲ ❛ ✇ ♦ ❢ ❚ ♦ ❛ ❧
❊ ① ♣ ❡ ❝ ❛ ✐ ♦ ♥ ❝ ♦ ♥ ❞ ✐ ✐ ♦ ♥ ❡ ❞ ♦ ♥ ❤ ❡ ❢ ❛ ❝ ❤ ❛ X An = xA ❛ ♥ ❞ X Bn = xB ✳ ■ ♥ ❡ ✉ ❛ ✐ ♦ ♥ ✸ ✹ ✇ ❡ ❝ ♦ ✉ ❧ ❞ ❡ ♠ ♦ ✈ ❡ ❤ ❡ ❝ ♦ ♥ ❞ ✐ ✐ ♦ ♥ ❜ ❡ ❝ ❛ ✉ ❡
❤ ❡ ❛ ♥ ❞ ♦ ♠ ✈ ❛ ✐ ❛ ❜ ❧ ❡ Y (AA)n,kA
✱ Y (AB)n,kA
✱ Y (BA)n,kB
❛ ♥ ❞ Y (BB)n,kB
❛ ❡ ✐ ♥ ❞ ❡ ♣ ❡ ♥ ❞ ❡ ♥ ♦ ❢ ❤ ❡ ❛ ♥ ❞ ♦ ♠ ✈ ❛ ✐ ❛ ❜ ❧ ❡ X An ❛ ♥ ❞ X Bn ✳ ■ ♥ ❡ ✉ ❛ ✐ ♦ ♥
✸ ✺ ✇ ❡ ❝ ♦ ✉ ❧ ❞ ❡ ♣ ❡ ❛ ❡ ❤ ❡ ✇ ♦ ❡ ① ♣ ❡ ❝ ❛ ✐ ♦ ♥ ❜ ❡ ❝ ❛ ✉ ❡ ❤ ❡ ❛ ♥ ❞ ♦ ♠ ✈ ❛ ✐ ❛ ❜ ❧ ❡ Y (AA)n,kA
❛ ♥ ❞ Y (AB)n,kA
✐ ✐ ♥ ❞ ❡ ♣ ❡ ♥ ❞ ❡ ♥ ♦ ❢ ❛ ♥ ❞ ♦ ♠
✈ ❛ ✐ ❛ ❜ ❧ ❡ Y (BA)n,kB
❛ ♥ ❞ Y (BB)n,kB
✳ ❊ ✉ ❛ ✐ ♦ ♥ ✸ ✻ ❛ ♥ ❞ ✸ ✼ ✐ ❤ ❡ ❡ ✉ ❧ ♦ ❢ ❤ ❡ ❢ ❛ ❝ ❤ ❛ ❞ ✐ ✐ ❜ ✉ ✐ ♦ ♥ ✭ ❛ ♥ ❞ ❤ ❡ ♥ ❝ ❡ ❤ ❡ ❡ ① ♣ ❡ ❝ ❛ ✐ ♦ ♥ ✮ ♦ ❢
❤ ❡ ❛ ♥ ❞ ♦ ♠ ✈ ❛ ✐ ❛ ❜ ❧ ❡ Y (AA)n,kA
❛ ♥ ❞ Y (AB)n,kA
✭ Y (BA)n,kB
❛ ♥ ❞ Y (BB)n,kB
✮ ✐ ✐ ♥ ❞ ❡ ♣ ❡ ♥ ❞ ❡ ♥ ♦ ❢ ❤ ❡ ❡ ♣ ♦ ❝ ❤ n ❛ ♥ ❞ ❤ ❡ ❛ ❣ ❡ ♥ ♥ ✉ ♠ ❜ ❡ kA ✭ kB ✮ ✳
❋ ✐ ♥ ❛ ❧ ❧ ② ✇ ❡ ❤ ❛ ✈ ❡ ✱
P XAn+1X
Bn+1
(u, v) = P XAn X
BnP (A) (u, v) , P (B) (u, v) ✭ ✸ ✽ ✮
♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❤ ❛ ❤ ❡ ❞ ❡ ❝ ❡ ♥ ❞ ❡ ♥ ✇ ✐ ❧ ❧ ❣ ♦ ❡ ① ✐ ♥ ❝ ❛ ❢ ❡ n ❡ ♣ ♦ ❝ ❤ ✐ P XAn X
Bn
(0, 0)✳ ❚ ♦ ✜ ♥ ❞ P XAn X
Bn
(0, 0) ✇ ❡ ❝ ❛ ♥ ❞ ♦ ❤ ❡
❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ❡ ❝ ✉ ✐ ♦ ♥
P XAn X
Bn
(0, 0) = P XAn−1X
Bn−1
P (A) (0, 0) , P (B) (0, 0)
P XA
n−1XBn−1
(u, v) = P XAn−2X
Bn−2
P (A) (u, v) , P (B) (u, v)
P XA
n XBn
(0, 0) = P XAn−2X
Bn−2
P (A)
P (A) (0, 0) , P (B) (0, 0)
, P (B)
P (A) (0, 0) , P (B) (0, 0)
✳
✳
✳
✳
✳
✳
✳
✳
✳
P XA
n XB
n
(0, 0) = P XA
0 XB
0
(α, β )
✇ ❤ ❡ ❡ α ❛ ♥ ❞ β ❛ ❡ ♦ ♠ ❡ ❝ ♦ ♥ ❛ ♥ ♦ ❜ ❛ ✐ ♥ ❡ ❞ ❛ ❢ ❡ n ❝ ♦ ♠ ♣ ♦ ✐ ✐ ♦ ♥ ♦ ❢ ❢ ✉ ♥ ❝ ✐ ♦ ♥ P (A) (u, v) ❛ ♥ ❞ P (B) (u, v) ✳ ❆ ❤ ✐ ♣ ♦ ✐ ♥ ✐
✐ ✐ ♥ ❡ ❡ ✐ ♥ ❣ ♦ ♥ ♦ ❡ ❛ ✈ ❡ ② ✐ ♠ ♣ ♦ ❛ ♥ ❡ ❧ ❛ ✐ ♦ ♥ ✇ ❤ ✐ ❝ ❤ ✇ ❡ ✇ ✐ ❧ ❧ ✉ ❡ ✐ ♠ ♠ ❡ ❞ ✐ ❛ ❡ ❧ ② ✳
P XAn X
Bn
(0, 0) = P XA0 X
B0
(α, β ) ⇒ P XAn+1X
Bn+1
(0, 0) = P XA1 X
B1
(α, β ) ✭ ✸ ✾ ✮
❚ ❤ ✐ ✐ ❜ ❡ ❝ ❛ ✉ ❡ ♦ ❣ ❡ ❢ ♦ ♠ ❡ ♣ ♦ ❝ ❤ n + 1 ♦ ❡ ♣ ♦ ❝ ❤ 1 ✇ ❡ ❤ ❛ ✈ ❡ ♦ ❞ ♦ n ❝ ♦ ♠ ♣ ♦ ✐ ✐ ♦ ♥ ♦ ❢ ❤ ❡ ❛ ♠ ❡ ❢ ✉ ♥ ❝ ✐ ♦ ♥ P (A) (u, v) ❛ ♥ ❞
P (B) (u, v) ❧ ✐ ❦ ❡ ✇ ❡ ❞ ✐ ❞ ♦ ❣ ❡ ❢ ♦ ♠ ❡ ♣ ♦ ❝ ❤ n ♦ ❡ ♣ ♦ ❝ ❤ 0 ✳ ▲ ❡ ❞ ❡ ✜ ♥ ❡ ✇ ♦ ✈ ❛ ✐ ❛ ❜ ❧ ❡
an10 = P
X An = 0 , X Bn = 0|X A0 = 1, X B0 = 0
✭ ✹ ✵ ✮
an01 = P
X An = 0 , X Bn = 0|X A0 = 0, X B0 = 1
✭ ✹ ✶ ✮
◗ ✉ ❛ ❧ ✐ ❛ ✐ ✈ ❡ ❧ ② ✱ an10 ✭ an01 ✮ ✐ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❤ ❛ ❤ ❡ ❞ ❡ ❝ ❡ ♥ ❞ ❛ ♥ ♣ ♦ ❞ ✉ ❝ ❡ ❞ ❜ ② ❛ ✐ ♥ ❣ ❧ ❡ ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ✭ ❚ ② ♣ ❡ ✲ ❇ ✮ ✇ ✐ ❧ ❧ ❜ ❡
❡ ① ✐ ♥ ❝ ❛ ❢ ❡ n ❡ ♣ ♦ ❝ ❤ ✳ ❧ ❡ ❛ ❡ ♥ ♦ ❡ ❤ ❛ an10 ❛ ♥ ❞ an01 ❛ ❡ ❡ ① ❛ ❝ ❧ ② ❤ ❡ ✉ ❛ ♥ ✐ ✐ ❡ ❛ ❦ ❡ ❞ ✐ ♥ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ✳ ◆ ♦ ✇ ✐ ❢
X A0 = 1 ❛ ♥ ❞ X B0 = 0 ✱ ✇ ❡ ❤ ❛ ✈ ❡
an10 = P XAn X
Bn
(0, 0)
= P XA0 X
B0
(α, β )
=xB
xA
αxAβ xBP
X A0 = xA, X B0 = xB
= α1β 0P
X A0 = 1, X B0 = 0
= α1β 0 · 1 = α ✭ ✹ ✷ ✮
an01 = β ♦ ♦ ❢ ✐ ✐ ♠ ✐ ❧ ❛ ♦ an10 = α ✭ ✹ ✸ ✮
❙ ✉ ❜ ✐ ✉ ✐ ♥ ❣
α❛ ♥ ❞
β ❢ ♦ ♠ ❡ ✉ ❛ ✐ ♦ ♥ ✹ ✷ ❛ ♥ ❞ ✹ ✸ ✐ ♥ ❡ ✉ ❛ ✐ ♦ ♥ ✸ ✾ ✇ ❡ ❣ ❡ ✱
P XAn+1X
Bn+1
(0, 0) = P XA1 X
B1
(an10, an01)
P XA1 X
B1
(u, v) = P XA0 X
B0
P (A) (u, v) , P (B) (u, v)
P XA
n+1XBn+1
(0, 0) = P XA0 X
B0
P (A) (an10, an01) , P (B) (an10, an01)
✭ ✹ ✹ ✮
❆ ♣ ♣ ❧ ② ✐ ♥ ❣ ❡ ✉ ❛ ✐ ♦ ♥ ✹ ✷ ❛ ♥ ❞ ✹ ✸ ✐ ♥ ❡ ✉ ❛ ✐ ♦ ♥ ✹ ✹ ✇ ❡ ❣ ❡
an+110 = P (A) (an10, an01) ✭ ✹ ✺ ✮
an+101 = P (B) (an10, an01) ✭ ✹ ✻ ✮
✶ ✸
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❊ ✉ ❛ ✐ ♦ ♥ ✹ ✺ ❛ ♥ ❞ ✹ ✻ ❞ ❡ ✜ ♥ ❡ ❛ ❡ ❝ ✉ ✐ ✈ ❡ ❡ ❧ ❛ ✐ ♦ ♥ ❤ ✐ ♣ ✇ ❤ ✐ ❝ ❤ ✇ ❡ ❝ ❛ ♥ ✉ ❡ ♦ ❣ ❡ an10 ❛ ♥ ❞ an01 ✳ ❍ ♦ ✇ ❡ ✈ ❡ ♦ ❛ ❤ ❡ ❡ ❝ ✉ ✐ ♦ ♥
✇ ❡ ♥ ❡ ❡ ❞ a010 ❛ ♥ ❞ a001 ✳ ■ ✐ ✐ ✈ ✐ ❛ ❧ ♦ ♦ ❜ ❡ ✈ ❡ ❤ ❛ a010 = 0 ✭ a001 = 0 ✮ ❛ X A0 = 1 ❛ ♥ ❞ X B0 = 0 ✭ X A0 = 0 ❛ ♥ ❞ X B0 = 1✮ ✳
❆ ♣ ♣ ❡ ♥ ❞ ✐ ① ✲ ❉ ❝ ♦ ♥ ❛ ✐ ♥ ❤ ❡ ▼ ❆ ❚ ▲ ❆ ❇ ❝ ♦ ❞ ❡ ✇ ❤ ✐ ❝ ❤ ✐ ♠ ♣ ❧ ❡ ♠ ❡ ♥ ❤ ✐ ❡ ❝ ✉ ✐ ♦ ♥ ♦ ❝ ❛ ❧ ❝ ✉ ❧ ❛ ❡ an10 ❛ ♥ ❞ an01 ✳
▲ ❊ ❆ ❙ ❊ ◆ ❖ ❚ ❊ ✿ ❚ ❤ ✐ ♣ ❛ ♦ ❢ ❤ ❡ ✉ ❡ ✐ ♦ ♥ ❛ ❧ ♦ ❛ ❦ ✉ ♦ ✏ ❝ ♦ ♠ ♣ ✉ ❡ ❤ ❡ ❡ ① ✐ ♥ ❝ ✐ ♦ ♥ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ✐ ❡ ❢ ♦ ❤ ❡
❡ ① ❛ ♠ ♣ ❧ ❡ ✇ ❡ ✐ ♠ ✉ ❧ ❛ ❡ ❞ ✐ ♥ ❛ ✭ ❜ ✮ ✑ ✳ ❲ ❡ ✇ ✐ ❧ ❧ ❞ ❡ ❢ ❡ ❤ ✐ ♦ ❛ ✭ ❜ ✮ ✇ ❤ ✐ ❝ ❤ ✇ ❡ ✇ ✐ ❧ ❧ ❛ ♥ ✇ ❡ ❜ ❡ ❧ ♦ ✇ ✳
✭ ❡ ✮ ▲ ❡ ✜ ❞ ❡ ✜ ♥ ❡ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ✈ ❛ ✐ ❛ ❜ ❧ ❡ ✱
a pq = P X An = 0 , X Bn = 0 ❢ ♦ ♦ ♠ ❡ n > 0|X A0 = p, X B0 = q ◗ ✉ ❛ ❧ ✐ ❛ ✐ ✈ ❡ ❧ ② ♣ ❡ ❛ ❦ ✐ ♥ ❣ ✱ a pq ✐ ❤ ❡ ❡ ① ✐ ♥ ❝ ✐ ♦ ♥ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❛ ✐ ♥ ❣ ❢ ♦ ♠ p ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ❛ ♥ ❞ q ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❇ ✳ ❚ ❤ ❡ ❦ ❡ ②
❞ ✐ ✛ ❡ ❡ ♥ ❝ ❡ ❜ ❡ ✇ ❡ ❡ ♥ a10 ✭ a01 ✮ ❛ ♥ ❞ an10 ✭ an01 ✮ ✐ ❤ ❛ a10 ✭ a01 ✮ ❞ ♦ ❡ ♥ ♦ ♣ ❡ ❝ ✐ ❢ ② ❤ ❡ ✐ ♠ ❡ ♦ ❢ ❡ ① ✐ ♥ ❝ ✐ ♦ ♥ ✳ ■ ♥ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ✇ ❡ ✇ ✐ ❧ ❧
✜ ❤ ♦ ✇ ❤ ❛ a pq = (a10) p
(a01)q
✳
a pq = P
pi=1
{❉ ❡ ❝ ❡ ♥ ❞ ❡ ♥ ♦ ❢ ✐ ♥ ✐ ✐ ❛ ❧ ❛ ❣ ❡ ♥ ★ ✐ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ❣ ♦ ❡ ① ✐ ♥ ❝ }
❛ ♥ ❞
qj=1
{❉ ❡ ❝ ❡ ♥ ❞ ❡ ♥ ♦ ❢ ✐ ♥ ✐ ✐ ❛ ❧ ❛ ❣ ❡ ♥ ★ ❥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❇ ❣ ♦ ❡ ① ✐ ♥ ❝ }
= P
p
i=1
{❉ ❡ ❝ ❡ ♥ ❞ ❡ ♥ ♦ ❢ ✐ ♥ ✐ ✐ ❛ ❧ ❛ ❣ ❡ ♥ ★ ✐ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ❣ ♦ ❡ ① ✐ ♥ ❝ }· P
qj=1
{❉ ❡ ❝ ❡ ♥ ❞ ❡ ♥ ♦ ❢ ✐ ♥ ✐ ✐ ❛ ❧ ❛ ❣ ❡ ♥ ★ ❥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❇ ❣ ♦ ❡ ① ✐ ♥ ❝ }
✭ ✹ ✼ ✮
=
pi=1
P ( ❉ ❡ ❝ ❡ ♥ ❞ ❡ ♥ ♦ ❢ ✐ ♥ ✐ ✐ ❛ ❧ ❛ ❣ ❡ ♥ ★ ✐ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ❣ ♦ ❡ ① ✐ ♥ ❝ )
·
qj=1
P ( ❉ ❡ ❝ ❡ ♥ ❞ ❡ ♥ ♦ ❢ ✐ ♥ ✐ ✐ ❛ ❧ ❛ ❣ ❡ ♥ ★ ❥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❇ ❣ ♦ ❡ ① ✐ ♥ ❝ ) ✭ ✹ ✽ ✮
=
p
i=1
a10
q
j=1
a01
= (a10) p
(a01)q
✭ ✹ ✾ ✮
❊ ✉ ❛ ✐ ♦ ♥ ✹ ✼ ❛ ♥ ❞ ✹ ✽ ✉ ❡ ✐ ♥ ❞ ❡ ♣ ❡ ♥ ❞ ❡ ♥ ❝ ❡ ✱ ♠ ♦ ❡ ♣ ❡ ❝ ✐ ❡ ❧ ② ✱ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❞ ❡ ❝ ❡ ♥ ❞ ❡ ♥ ♦ ❢ ♦ ♥ ❡ ❛ ❣ ❡ ♥ ✐ ✐ ♥ ❞ ❡ ♣ ❡ ♥ ❞ ❡ ♥ ♦ ❢ ❤ ❡
♥ ✉ ♠ ❜ ❡ ♦ ❢ ❞ ❡ ❝ ❡ ♥ ❞ ❡ ♥ ♦ ❢ ❛ ♥ ♦ ❤ ❡ ❛ ❣ ❡ ♥ ✳ ❊ ✉ ❛ ✐ ♦ ♥ ✹ ✾ ✉ ❡ ❤ ❡ ❢ ❛ ❝ ❤ ❛ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❞ ✐ ✐ ❜ ✉ ✐ ♦ ♥ ♦ ❢ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢
❞ ❡ ❝ ❡ ♥ ❞ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ✭ ❚ ② ♣ ❡ ✲ ❇ ✮ ❛ ❣ ❡ ♥ ❞ ♦ ❡ ♥ ♦ ✈ ❛ ② ✇ ✐ ❤ ❛ ❣ ❡ ♥ ♥ ✉ ♠ ❜ ❡ ✳ ◆ ♦ ✇ ❝ ♦ ♥ ✐ ❞ ❡ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ✱
a10 = P
∞n=1
X An = 0, X Bn = 0
|X A0 = 1, X B0 = 0
✭ ✺ ✵ ✮
=∞j=0
∞i=0
P
∞n=1
X An = 0, X Bn = 0
|X A1 = i, X B1 = j, X A0 = 1, X B0 = 0
· P X A1 = i, X B1 = j |X A0 = 1, X B0 = 0 ✭ ✺ ✶ ✮
=∞j=0
∞i=0
P
∞n=1
X An = 0, X Bn = 0
|X A1 = i, X B1 = j, X A0 = 1, X B0 = 0
p(A) (i, j)
=∞j=0
∞i=0
P
∞n=1
X An = 0, X Bn = 0
|X A1 = i, X B1 = j
p(A) (i, j)
✭ ✺ ✷ ✮
=∞j=0
∞i=0
aij p
(A) (i, j)
✭ ✺ ✸ ✮
=∞j=0
∞i=0
(a10)
i(a01)
j p(A) (i, j)
= P (A) (a10, a01) ✭ ✺ ✹ ✮
✶ ✹
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❚ ♦ ❣ ❡ ❡ ✉ ❛ ✐ ♦ ♥ ✺ ✶ ✱ ✇ ❡ ✉ ❡ ❤ ❡ ▲ ❛ ✇ ♦ ❢ ❚ ♦ ❛ ❧ ❊ ① ♣ ❡ ❝ ❛ ✐ ♦ ♥ ❝ ♦ ♥ ❞ ✐ ✐ ♦ ♥ ❡ ❞ ♦ ♥ ❤ ❡ ❢ ❛ ❝ ❤ ❛ X A1 = i ❛ ♥ ❞ X B1 = j ✳ ❊ ✉ ❛ ✐ ♦ ♥ ✺ ✷
✐ ❤ ❡ ❡ ✉ ❧ ♦ ❢ ❤ ❡ ▼ ❛ ❦ ♦ ✈ ♦ ♣ ❡ ② ✳ ❊ ✉ ❛ ✐ ♦ ♥ ✺ ✸ ✐ ♦ ❜ ❛ ✐ ♥ ❡ ❞ ❜ ② ✐ ♠ ❡ ❤ ✐ ❢ ✐ ♥ ❣ n → n + 1 ✳ ❊ ✉ ❛ ✐ ♦ ♥ ✺ ✹ ✐ ♦ ❜ ❛ ✐ ♥ ❡ ❞ ❜ ②
✉ ✐ ♥ ❣ ❡ ✉ ❛ ✐ ♦ ♥ ✹ ✾ ✳ ❙ ✐ ♠ ✐ ❧ ❛ ❧ ② ✇ ❡ ❝ ❛ ♥ ♣ ♦ ✈ ❡ ❤ ❛ a01 = P (B) (a10, a01)✳ ❲ ❡ ✜ ♥ ❛ ❧ ❧ ② ❤ ❛ ✈ ❡ ✱
a10 = P (A) (a10, a01) ✭ ✺ ✺ ✮
a01 = P (B) (a10, a01) ✭ ✺ ✻ ✮
❊ ✉ ❛ ✐ ♦ ♥ ✺ ✺ ❛ ♥ ❞ ✺ ✻ ✐ ❛ ② ❡ ♠ ♦ ❢ ♥ ♦ ♥ ❧ ✐ ♥ ❡ ❛ ❡ ✉ ❛ ✐ ♦ ♥ ✇ ✐ ❤ ✇ ♦ ✈ ❛ ✐ ❛ ❜ ❧ ❡ ✇ ❤ ✐ ❝ ❤ ❝ ❛ ♥ ❜ ❡ ♦ ❧ ✈ ❡ ❞ ♦ ♦ ❜ ❛ ✐ ♥ a10 ❛ ♥ ❞ a01 ✳
❧ ❡ ❛ ❡ ♥ ♦ ❡ ❤ ❛ a10 ❛ ♥ ❞ a01 ❛ ❡ ❡ ① ❛ ❝ ❧ ② ❤ ❡ ✉ ❛ ♥ ✐ ✐ ❡ ❛ ❦ ❡ ❞ ✐ ♥ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ✳ ❖ ♥ ❡ ✇ ❛ ② ♦ ♦ ❧ ✈ ❡ ❡ ✉ ❛ ✐ ♦ ♥ ✺ ✺
❛ ♥ ❞ ✺ ✻ ✐ ♦ ✉ ❡ ❤ ❡ ♦ ❧ ✈ ❡ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ♦ ❢ ▼ ❛ ❤ ❡ ♠ ❛ ✐ ❝ ❛ ✳ ❍ ♦ ✇ ❡ ✈ ❡ ❤ ❡ ❡ ② ❡ ♠ ♦ ❢ ❡ ✉ ❛ ✐ ♦ ♥ ♠ ❛ ② ❤ ❛ ✈ ❡ ♠ ✉ ❧ ✐ ♣ ❧ ❡ ♦ ❧ ✉ ✐ ♦ ♥ ✳ ❖ ♥ ❡
✐ ✈ ✐ ❛ ❧ ♦ ❧ ✉ ✐ ♦ ♥ ✐ a10 = a01 = 1 ✳ ❚ ❤ ✐ ✐ ❜ ❡ ❝ ❛ ✉ ❡ P (A) (1, 1) = P (B) (1, 1) = 1 ✳ ❚ ❤ ❡ ❡ ❝ ❛ ♥ ❜ ❡ ♦ ❤ ❡ ♥ ♦ ♥ ✲ ✐ ✈ ✐ ❛ ❧ ♦ ❧ ✉ ✐ ♦ ♥
♦ ♦ ✳ ● ✐ ✈ ❡ ♥ ❤ ❛ ✐ ✐ ❛ ② ❡ ♠ ♦ ❢ ♥ ♦ ♥ ❧ ✐ ♥ ❡ ❛ ❡ ✉ ❛ ✐ ♦ ♥ ✐ ♥ ✇ ♦ ✈ ❛ ✐ ❛ ❜ ❧ ❡ ✱ ✐ ✐ ❞ ✐ ✣ ❝ ✉ ❧ ♦ ✉ ♥ ❞ ❡ ❛ ♥ ❞ ❤ ❡ ❣ ❡ ♦ ♠ ❡ ② ♦ ❢ ❤ ❡
❛ ♦ ❝ ✐ ❛ ❡ ❞ ❣ ❛ ♣ ❤ ❡ ✉ ✐ ❡ ❞ ♦ ♦ ❧ ✈ ❡ ❤ ❡ ♣ ♦ ❜ ❧ ❡ ♠ ✳ ■ ♥ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ✇ ❡ ✇ ✐ ❧ ❧ ✉ ❣ ❣ ❡ ❛ ♥ ♦ ❤ ❡ ✇ ❛ ② ♦ ♦ ❧ ✈ ❡ ❡ ✉ ❛ ✐ ♦ ♥ ✺ ✺ ❛ ♥ ❞
✺ ✻ ❜ ② ❞ ❡ ✈ ❡ ❧ ♦ ♣ ✐ ♥ ❣ ❛ ❡ ❧ ❛ ✐ ♦ ♥ ❜ ❡ ✇ ❡ ❡ ♥ a10 ❛ ♥ ❞ an10 ✭ a01 ❛ ♥ ❞ an01 ✮ ✳
❊ ✉ ❛ ✐ ♦ ♥ ✺ ✵ ❝ ❛ ♥ ❛ ❧ ♦ ❜ ❡ ❡ ① ♣ ❡ ❡ ❞ ❛
a10 = limN →∞
P
N n=1
X An = 0, X Bn = 0
|X A0 = 1, X B0 = 0
✭ ✺ ✼ ✮
■ ❢ X A
n
= 0 ❛ ♥ ❞ X B
n
= 0 ❢ ♦ ❛ ❣ ✐ ✈ ❡ ♥ n✱ ❤ ❡ ♥ ❢ ♦ ❛ ♥ ② n ≥ n ✇ ❡ ❝ ❛ ♥ ✐ ♠ ♠ ❡ ❞ ✐ ❛ ❡ ❧ ② ❛ ② X A
n
= 0 ❛ ♥ ❞ X B
n
= 0 ✳ ❚ ❤ ✐ ✐ ❜ ❡ ❝ ❛ ✉ ❡
(0, 0) ✐ ❛ ♥ ❛ ❜ ♦ ❜ ✐ ♥ ❣ ❛ ❡ ✳ ❚ ❤ ✐ ✐ ♠ ❛ ❤ ❡ ♠ ❛ ✐ ❝ ❛ ❧ ❧ ② ❡ ① ♣ ❡ ❡ ❞ ❛
X An = 0, X Bn = 0
⊆
X An = 0, X Bn = 0
; n ≥ n
⇒N n=1
X An = 0, X Bn = 0
=
X AN = 0, X BN = 0
✭ ✺ ✽ ✮
❙ ✉ ❜ ✐ ✉ ✐ ♥ ❣ ❡ ✉ ❛ ✐ ♦ ♥ ✺ ✽ ✐ ♥ ❡ ✉ ❛ ✐ ♦ ♥ ✺ ✼ ✇ ❡ ❣ ❡
a10 = limN →∞
P
X AN = 0, X BN = 0
|X A0 = 1, X B0 = 0
= limN →∞ aN
10✭ ✺ ✾ ✮
❊ ✉ ❛ ✐ ♦ ♥ ✺ ✾ ✐ ♦ ❜ ❛ ✐ ♥ ❡ ❞ ❢ ♦ ♠ ❤ ❡ ❞ ❡ ✜ ♥ ✐ ✐ ♦ ♥ ♦ ❢ an10 ✭ ❡ ❢ ❡ ❡ ✉ ❛ ✐ ♦ ♥ ✹ ✵ ✮ ✳ ❙ ✐ ♠ ✐ ❧ ❛ ❧ ② ✇ ❡ ❝ ❛ ♥ ♣ ♦ ✈ ❡ ❤ ❛
a01 = limN →∞
aN 01 ✭ ✻ ✵ ✮
❊ ✉ ❛ ✐ ♦ ♥ ✺ ✾ ❛ ♥ ❞ ✻ ✵ ✉ ❣ ❣ ❡ ❤ ❛ ✇ ❡ ❝ ❛ ♥ ✜ ♥ ❞ a10 ❛ ♥ ❞ a01 ❜ ② ✐ ♠ ♣ ❧ ② ❝ ❛ ❧ ❝ ✉ ❧ ❛ ✐ ♥ ❣ aN 10 ❛ ♥ ❞ aN 01 ❡ ♣ ❡ ❝ ✐ ✈ ❡ ❧ ② ❢ ♦ ❧ ❛ ❣ ❡ n✳ ❲ ❡
❝ ❛ ♥ ❞ ♦ ❤ ✐ ❜ ② ✉ ✐ ♥ ❣ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ❡ ❝ ✉ ✐ ✈ ❡ ❡ ❧ ❛ ✐ ♦ ♥ ❞ ❡ ✈ ❡ ❧ ♦ ♣ ❡ ❞ ✐ ♥ ❛ ✭ ❞ ✮
an+110 = P (A) (an10, an01) ✭ ✻ ✶ ✮
an+101 = P (B) (an10, an01) ✭ ✻ ✷ ✮
❖ ❜ ❡ ✈ ❡ ❤ ❛ ❡ ✉ ❛ ✐ ♦ ♥ ✻ ✶ ❛ ♥ ❞ ✻ ✷ ❞ ❡ ✜ ♥ ❡ ❛ ♥ ♦ ♥ ❧ ✐ ♥ ❡ ❛ ❞ ✐ ❝ ❡ ❡ ✐ ♠ ❡ ❞ ② ♥ ❛ ♠ ✐ ❝ ❛ ❧ ② ❡ ♠ ✇ ✐ ❤ ❛ ❡ (an10, an01)✳ ❖ ♥ ❡ ❝ ❛ ♥
✉ ❡ ✐ ♦ ♥ ❤ ❡ ❛ ❜ ✐ ❧ ✐ ② ❛ ♥ ❞ ❝ ♦ ♥ ✈ ❡ ❣ ❡ ♥ ❝ ❡ ♦ ❢ ❤ ✐ ❞ ② ♥ ❛ ♠ ✐ ❝ ❛ ❧ ② ❡ ♠ ✳ ❚ ♦ ♣ ♦ ✈ ❡ ❛ ❜ ✐ ❧ ✐ ② ✇ ❡ ❝ ❛ ♥ ❤ ♦ ✇ ❤ ❛ ❤ ❡ ❡
S = {(x, y) : 0 ≤ x ≤ 1 ; 0 ≤ y ≤ 1}
✐ ✐ ♥ ✈ ❛ ✐ ❛ ♥ ✳ ❚ ♦ ♣ ♦ ✈ ❡ ❤ ✐ ❝ ♦ ♥ ✐ ❞ ❡ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ❛ ❣ ✉ ♠ ❡ ♥ ✳ ❋ ✐ ♥ ♦ ❡ ❤ ❛ a010 = 0 ❛ ♥ ❞ a001 = 0 ✱ ✐ ✳ ❡
a010, a001
∈ S ✳ ❙ ❛ ②
❤ ❛ (an10, an01) ∈ S ✳ ◆ ♦ ✇
an+110 = P (A) (an10, an01)
= E
(an10)Y (AA)
(an01)Y (AB)
✭ ✻ ✸ ✮
✶ ✺
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◆ ♦ ✇ 0 ≤ an10 ≤ 1 ❛ ♥ ❞ 0 ≤ an01 ≤ 1 ✐ ♥ ❝ ❡ (an10, an01) ∈ S ✳ ❆ ❧ ♦ Y (AA) ≥ 0 ❛ ♥ ❞ Y (AB) ≥ 0 ✳ ❚ ❤ ✐ ✐ ♠ ♣ ❧ ✐ ❡ ❤ ❛
0 ≤ (an10)Y (AA)
· (an01)Y (AB)
≤ 1 ❛ ♥ ❞ ❤ ❡ ♥ ❝ ❡ 0 ≤ an+110 = E
(an10)
Y (AA)
(an01)Y (AB)
≤ 1 ✳ ❙ ✐ ♠ ✐ ❧ ❛ ❧ ② ✇ ❡ ❝ ❛ ♥ ♣ ♦ ✈ ❡ ❤ ❛
0 ≤ an+101 ≤ 1 ✳ ❍ ❡ ♥ ❝ ❡
an+110 , an+1
01
∈ S ✳ ❲ ❡ ✜ ♥ ❛ ❧ ❧ ② ❝ ♦ ♥ ❝ ❧ ✉ ❞ ❡ ❤ ❛ ✐ ❢
a010, a001
∈ S ❤ ❡ ♥ (an10, an01) ∈ S ; ∀n ≥ 0✳ ❚ ❤ ✐
♣ ♦ ✈ ❡ ❤ ❛ S ✐ ❛ ♥ ✐ ♥ ✈ ❛ ✐ ❛ ♥ ❡ ❛ ♥ ❞ ❤ ❡ ♥ ❝ ❡ an10 ❛ ♥ ❞ an01 ✐ ❧ ♦ ✇ ❡ ❛ ♥ ❞ ✉ ♣ ♣ ❡ ❜ ♦ ✉ ♥ ❞ ❡ ❞ ❜ ② 0 ❛ ♥ ❞ 1 ❡ ♣ ❡ ❝ ✐ ✈ ❡ ❧ ② ✳ ❚ ❤ ✐ ❝ ♦ ♠ ♣ ❧ ❡ ❡
❤ ❡ ♣ ♦ ♦ ❢ ❤ ❛ ❤ ❡ ❞ ② ♥ ❛ ♠ ✐ ❝ ❛ ❧ ② ❡ ♠ ❞ ❡ ✜ ♥ ❡ ❞ ❜ ② ❡ ✉ ❛ ✐ ♦ ♥ ✻ ✶ ❛ ♥ ❞ ✻ ✷ ✐ ❛ ❜ ❧ ❡ ✳
❍ ♦ ✇ ❡ ✈ ❡ ✱ ❡ ✈ ❡ ♥ ✐ ❢ ❤ ❡ ❞ ② ♥ ❛ ♠ ✐ ❝ ❛ ❧ ② ❡ ♠ ✐ ❛ ❜ ❧ ❡ ✐ ♠ ❛ ② ♥ ♦ ❝ ♦ ♥ ✈ ❡ ❣ ❡ ✳ ❘ ❛ ❤ ❡ ✱ ✐ ♠ ❛ ② ❣ ♦ ✐ ♥ ♦ ❛ ❧ ✐ ♠ ✐ ❝ ② ❝ ❧ ❡ ✳ ■ ❤ ♦ ✉ ❧ ❞ ❜ ❡
♣ ♦ ✐ ❜ ❧ ❡ ♦ ♣ ♦ ✈ ❡ ❤ ❛ ❤ ✐ ❞ ② ♥ ❛ ♠ ✐ ❝ ❛ ❧ ② ❡ ♠ ❞ ♦ ❡ ❝ ♦ ♥ ✈ ❡ ❣ ❡ ✱ ❜ ✉ ✇ ❡ ❝ ♦ ✉ ❧ ❞ ♥ ♦ ♣ ♦ ✈ ❡ ✐ ✳
❚ ❤ ❡ ▼ ❆ ❚ ▲ ❆ ❇ ❝ ♦ ❞ ❡ ♦ ❝ ❛ ❧ ❝ ✉ ❧ ❛ ❡ a10 ❛ ♥ ❞ a01 ✐ ✐ ♥ ❝ ❧ ✉ ❞ ❡ ❞ ✐ ♥ ❆ ♣ ♣ ❡ ♥ ❞ ✐ ① ✲ ❉ ✳
✭ ❢ ✮ ■ ♥ ▲ ❡ ❝ ✉ ❡ ✶ ✻ ✱ ✇ ❡ ❧ ❡ ❛ ♥ ❡ ❞ ❛ ❜ ♦ ✉ ❤ ♦ ✇ ♦ ✐ ♥ ❢ ❡ ✱ ✐ ❢ ❛ ❜ ❛ ♥ ❝ ❤ ✐ ♥ ❣ ♣ ♦ ❝ ❡ ✐ ❣ ♦ ✐ ♥ ❣ ♦ ❡ ① ✐ ♥ ❝ ✱ ❜ ② ❝ ❤ ❡ ❝ ❦ ✐ ♥ ❣ ❤ ❡ ❡ ① ♣ ❡ ❝ ❛ ♥ ❝ ②
♦ ❢ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ♦ ✛ ♣ ✐ ♥ ❣ ✳ ❲ ❡ ✇ ♦ ✉ ❧ ❞ ❧ ✐ ❦ ❡ ♦ ❞ ♦ ❤ ❡ ❛ ♠ ❡ ❤ ❡ ❡ ✳ ❙ ♦ ✇ ❡ ❝ ❛ ❧ ❝ ✉ ❧ ❛ ❡ ❤ ❡ ❡ ① ♣ ❡ ❝ ❛ ♥ ❝ ② ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ❛ ♥ ❞ ❚ ② ♣ ❡ ✲ ❇
❛ ❣ ❡ ♥ ❛ ❤ ❡ nth ❡ ♣ ♦ ❝ ❤
E
X An+1
=
u
d
du
P XA
n+1XBn+1
(u, 1)u=1
= u d
du
P XA
n XBn
P (A) (u, 1) P (A) (1, 1) , P (B) (u, 1) P (B) (1, 1)
u=1
✭ ✻ ✹ ✮
= u
d
du P XAn X
Bn P
(A)
(u, 1) , P
(B)
(u, 1)u=1
= u d
du
P XA
n XBn
(X (u) , Y (u))u=1
✇ ❤ ❡ ❡ X (u) = P (A) (u, 1) , Y (u) = P (B) (u, 1)
=
u
dX
du · P X XA
n XBn
(X (u) , Y (u)) + udY
du · P Y
XAn X
Bn
(X (u) , Y (u))
u=1
✭ ✻ ✺ ✮
= E
Y (AA)
· P X XAn X
Bn
(1, 1) + E
Y (BA)
· P Y XAn X
Bn
(1, 1)
= E
Y (AA)
· E
X An
+ E
Y (BA)
· E
X Bn
✭ ✻ ✻ ✮
❊ ✉ ❛ ✐ ♦ ♥ ✻ ✹ ✐ ♦ ❜ ❛ ✐ ♥ ❡ ❞ ❜ ② ✉ ❜ ✐ ✉ ✐ ♥ ❣ (u, v) = (u, 1) ✐ ♥ ❡ ✉ ❛ ✐ ♦ ♥ ✸ ✽ ✳ ■ ♥ ❡ ✉ ❛ ✐ ♦ ♥ ✻ ✺ P X XAn X
Bn
(X (u) , Y (u)) ✭ P Y XAn X
Bn
(X (u) ,
♠ ❡ ❛ ♥ ♣ ❛ ✐ ❛ ❧ ❞ ✐ ✛ ❡ ❡ ♥ ✐ ❛ ✐ ♦ ♥ ♦ ❢ P XAn X
Bn
✇ ✐ ❤ ❡ ♣ ❡ ❝ ♦ X ✭ Y ✮ ✳ ❙ ✐ ♠ ✐ ❧ ❛ ❧ ② ✇ ❡ ❝ ❛ ♥ ♣ ♦ ✈ ❡
E X Bn+1 = E Y (AB)E X An + E Y (BB)E X Bn ✭ ✻ ✼ ✮
❈ ♦ ♠ ❜ ✐ ♥ ✐ ♥ ❣ ❡ ✉ ❛ ✐ ♦ ♥ ✻ ✻ ❛ ♥ ❞ ✻ ✼ ✇ ❡ ❣ ❡ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ♠ ❛ ✐ ① ❡ ✉ ❛ ✐ ♦ ♥ E
X An+1
E
X Bn+1
=
E
Y (AA)
E
Y (BA)
E
Y (AB)
E
Y (BB) E
X An
E
X Bn
▲ ❡ µAA = E
Y (AA)
✱ µAB = E
Y (AB)
✱ µBA = E
Y (BA)
❛ ♥ ❞ µBB = E
Y (BB)
✳ ❙ ✉ ❜ ✐ ✉ ✐ ♥ ❣ ✐ ♥ ❤ ❡ ❛ ❜ ♦ ✈ ❡ ❡ ✉ ❛ ✐ ♦ ♥
✇ ❡ ❣ ❡ E
X An+1
E
X Bn+1
=
µAA µBA
µAB µBB
E
X An
E
X Bn
✭ ✻ ✽ ✮
❊ ✉ ❛ ✐ ♦ ♥ ✻ ✽ ❞ ❡ ✜ ♥ ❡ ❛ ▲ ❚ ■ ② ❡ ♠ ✇ ✐ ❤ ✇ ♦ ❛ ❡ ✳ ❚ ❤ ❡ ❡ ✐ ❣ ❡ ♥ ✈ ❛ ❧ ✉ ❡ ♦ ❢ ❤ ❡ ② ❡ ♠ ❛ ❡
λM =
µAA + µBB
+
(µAA − µBB)
2+ 4µBAµAB
2✭ ✻ ✾ ✮
λm =
µAA + µBB
−
(µAA − µBB)
2+ 4µBAµAB
2✭ ✼ ✵ ✮
λM ❛ ♥ ❞ λm ❞ ❡ ✜ ♥ ❡ ❞ ❜ ② ❡ ✉ ❛ ✐ ♦ ♥ ✻ ✾ ❛ ♥ ❞ ✼ ✵ ❤ ❛ ✈ ❡ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ♣ ♦ ♣ ❡ ✐ ❡ ✿
✶ ✮ λM ❛ ♥ ❞ λm ❛ ❡ ❛ ❧ ✇ ❛ ② ❡ ❛ ❧ ✳
✷ ✮ λM ≥ 0✸ ✮ λM ≥ |λm| ✳
✶ ✻
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▲ ❡ X A0 = p ❛ ♥ ❞ X B0 = q ✐ ♠ ♣ ❧ ② ✐ ♥ ❣ E
X A0
= p ❛ ♥ ❞ E
X B0
= q ✳ ❖ ♥ ❡ ❝ ❛ ♥ ❤ ♦ ✇ ❤ ❛ ❤ ❡ ❣ ❡ ♥ ❡ ❛ ❧ ✉ ❝ ✉ ❡ ♦ ❢ ❤ ❡ ✐ ♠ ❡
❡ ✈ ♦ ❧ ✉ ✐ ♦ ♥ ♦ ❢ E
X An
❛ ♥ ❞ E
X Bn
✐
E
X An
= α1 (λM )n
p + α2 (λm)n
q ✭ ✼ ✶ ✮
E
X Bn
= β 1 (λM )n
p + β 2 (λm)n
q ✭ ✼ ✷ ✮
❚ ❛ ❦ ✐ ♥ ❣ ❤ ❡ ❧ ♣ ❢ ♦ ♠ ❧ ❡ ❝ ✉ ❡ ✶ ✻ ✭ ♣ ❛ ❣ ❡ ✸ ✮ ✇ ❡ ❞ ❡ ✈ ❡ ❧ ♦ ♣ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ✉ ❧ ❡ ✿
✶ ✮ ■ ❢ limn→∞
E X An = 0 ✭ limn→∞
E X Bn = 0 ✮ ❤ ❡ ♥ ❚ ② ♣ ❡ ✲ ❆ ❛ ❣ ❡ ♥ ✭ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ✮ ❲ ■ ▲ ▲ ❜ ❡ ❝ ♦ ♠ ❡ ❡ ① ✐ ♥ ❝ ✳
✷ ✮ ■ ❢ limn→∞
E
X An
= θ ✭ limn→∞
E
X Bn
= θ ✮ ✱ ✇ ❤ ❡ ❡ θ ✐ ❛ ✜ ♥ ✐ ❡ ❝ ❛ ❧ ❛ ✉ ❛ ♥ ✐ ② ✱ ❤ ❡ ♥ ❚ ② ♣ ❡ ✲ ❆ ❛ ❣ ❡ ♥ ✭ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ✮ ❲ ■ ▲ ▲
❜ ❡ ❝ ♦ ♠ ❡ ❡ ① ✐ ♥ ❝ ✳
✸ ✮ ■ ❢ limn→∞
E
X An
= ∞ ✭ limn→∞
E
X Bn
= ∞ ✮ ❤ ❡ ♥ ❚ ② ♣ ❡ ✲ ❆ ❛ ❣ ❡ ♥ ✭ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ✮ ▼ ❆ ❨ ❜ ❡ ❝ ♦ ♠ ❡ ❡ ① ✐ ♥ ❝ ✳ ■ ✐ ✏ ▼ ❆ ❨ ✑ ❜ ❡ ❝ ❛ ✉ ❡
✐ ♥ ❤ ✐ ❝ ❛ ❡ ❤ ❡ ❡ ✐ ❛ ✜ ♥ ✐ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ♦ ❢ ❡ ① ✐ ♥ ❝ ✐ ♦ ♥ ✳
❈ ♦ ♥ ❞ ✐ ✐ ♦ ♥ ❢ ♦ ✇ ❤ ✐ ❝ ❤ ❜ ♦ ❤ ❤ ❡ ❛ ❣ ❡ ♥ ❲ ■ ▲ ▲ ❜ ❡ ❝ ♦ ♠ ❡ ❡ ① ✐ ♥ ❝ ❢ ♦ ❛ ♥ ② ✐ ♥ ✐ ✐ ❛ ❧ ♣ ♦ ♣ ✉ ❧ ❛ ✐ ♦ ♥ ✿
❚ ❤ ❡ ❝ ♦ ♥ ❞ ✐ ✐ ♦ ♥ ❢ ♦ ❡ ① ✐ ♥ ❝ ✐ ♦ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ❛ ♥ ❞ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ❢ ♦ ❛ ♥ ② ✐ ♥ ✐ ✐ ❛ ❧ ♣ ♦ ♣ ✉ ❧ ❛ ✐ ♦ ♥
λM ≤ 1 ⇒µAA + µBB+ (µAA − µBB)2 + 4µBAµAB
2 ≤ 1 ✭ ✼ ✸ ✮
■ ❢ λM < 1 ❤ ❡ ♥ ❢ ♦ ♠ ❡ ✉ ❛ ✐ ♦ ♥ ✼ ✶ ❛ ♥ ❞ ✼ ✷ ✇ ❡ ❝ ❛ ♥ ❡ ❡ ❤ ❛ limn→∞
E
X An
= 0 ❛ ♥ ❞ limn→∞
E
X Bn
= 0✳ ❍ ❡ ♥ ❝ ❡ ❜ ♦ ❤ ❚ ② ♣ ❡ ✲ ❆ ❛ ♥ ❞
❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ✇ ✐ ❧ ❧ ❜ ❡ ❝ ♦ ♠ ❡ ❡ ① ✐ ♥ ❝ ✳
■ ❢ λM = 1 ❤ ❡ ♥ limn→∞
E
X An
= θ ❛ ♥ ❞ limn→∞
E
X Bn
= φ ✱ ✇ ❤ ❡ ❡ θ ❛ ♥ ❞ φ ❛ ❡ ✜ ♥ ✐ ❡ ❝ ❛ ❧ ❛ ✉ ❛ ♥ ✐ ② ✳ ❍ ❡ ♥ ❝ ❡ ❜ ♦ ❤ ❚ ② ♣ ❡ ✲ ❆ ❛ ♥ ❞
❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ✇ ✐ ❧ ❧ ❜ ❡ ❝ ♦ ♠ ❡ ❡ ① ✐ ♥ ❝ ✳
❧ ❡ ❛ ❡ ♥ ♦ ❡ ❤ ❛ ✐ ♥ ❡ ✉ ❛ ❧ ✐ ② ✼ ✸ ✐ ❤ ❡ ❝ ✐ ❡ ✐ ❛ ❢ ♦ ❡ ① ✐ ♥ ❝ ✐ ♦ ♥ ✇ ❤ ✐ ❝ ❤ ✐ ❛ ❦ ❡ ❞ ✐ ♥ ❛ ✭ ❢ ✮ ✳ ❲ ❡ ❛ ❧ ♦ ❞ ❡ ✐ ✈ ❡ ❢ ❡ ✇
♦ ❤ ❡ ❝ ✐ ❡ ✐ ❛ ❜ ❡ ❧ ♦ ✇ ♦ ❤ ❡ ❧ ♣ ✉ ✇ ✐ ❤ ❤ ❡ ▼ ♦ ♥ ❡ ❈ ❛ ❧ ♦ ✐ ♠ ✉ ❧ ❛ ✐ ♦ ♥ ✐ ♥ ❛ ✭ ❜ ✮ ✳
❈ ♦ ♥ ❞ ✐ ✐ ♦ ♥ ❢ ♦ ✇ ❤ ✐ ❝ ❤ ❜ ♦ ❤ ❤ ❡ ❛ ❣ ❡ ♥ ▼ ❆ ❨ ✢ ♦ ✉ ✐ ❤ ❢ ♦ ❛ ♥ ② ✐ ♥ ✐ ✐ ❛ ❧ ♣ ♦ ♣ ✉ ❧ ❛ ✐ ♦ ♥ ✿
❈ ♦ ♥ ❞ ✐ ✐ ♦ ♥ ❤ ❛ ❜ ♦ ❤ ❛ ❣ ❡ ♥ ▼ ❆ ❨ ✢ ♦ ✉ ✐ ❤ ❢ ♦ ❛ ♥ ② ✐ ♥ ✐ ✐ ❛ ❧ ♣ ♦ ♣ ✉ ❧ ❛ ✐ ♦ ♥ ✐
λM > 1 or |λm| > 1 ✭ ✼ ✹ ✮
❯ ♥ ❞ ❡ ❛ ♥ ② ♦ ❢ ❤ ❡ ❡ ❝ ♦ ♥ ❞ ✐ ✐ ♦ ♥ limn→∞
E
X An
= ∞ ❛ ♥ ❞ limn→∞
E
X Bn
= ∞ ✳ ❍ ❡ ♥ ❝ ❡ ❜ ♦ ❤ ❛ ❣ ❡ ♥ ♠ ❛ ② ✢ ♦ ✉ ✐ ❤ ✳
❈ ♦ ♥ ❞ ✐ ✐ ♦ ♥ ❢ ♦ ✇ ❤ ✐ ❝ ❤ ❚ ② ♣ ❡ ✲ ❆ ❛ ❣ ❡ ♥ ❲ ■ ▲ ▲ ❡ ① ✐ ♥ ❝ ❜ ✉ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ▼ ❆ ❨ ✢ ♦ ✉ ✐ ❤ ❢ ♦ ❛ ♥ ② ✐ ♥ ✐ ✐ ❛ ❧ ♣ ♦ ♣ ✉ ❧ ❛ ✐ ♦
❈ ♦ ♥ ❞ ✐ ✐ ♦ ♥ ❤ ❛ ❚ ② ♣ ❡ ✲ ❆ ❛ ❣ ❡ ♥ ✇ ✐ ❧ ❧ ❡ ① ✐ ♥ ❝ ❛ ♥ ❞ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ♠ ❛ ② ✢ ♦ ✉ ✐ ❤ ✐
µBA = 0 and µAA ≤ 1 and µBB > 1 ✭ ✼ ✺ ✮
❇ ② ❛ ♣ ♣ ❧ ② ✐ ♥ ❣ ❤ ❡ ❡ ❝ ♦ ♥ ❞ ✐ ✐ ♦ ♥ ✐ ♥ ❡ ✉ ❛ ✐ ♦ ♥ ✻ ✽ ✇ ❡ ❝ ❛ ♥ ❝ ♦ ♥ ❝ ❧ ✉ ❞ ❡ ❤ ❛ ✿
✶ ✮ limn→∞
E
X An
= 0 ✐ ❢ µAA < 1 ✐ ♠ ♣ ❧ ② ✐ ♥ ❣ ❤ ❛ ❚ ② ♣ ❡ ✲ ❆ ❛ ❣ ❡ ♥ ✇ ✐ ❧ ❧ ❜ ❡ ❝ ♦ ♠ ❡ ❡ ① ✐ ♥ ❝ ✳ ■ ❢ µAA = 1 ❤ ❡ ♥ limn→∞
E
X An
= θ ❛ ❣ ❛ ✐ ♥
✐ ♠ ♣ ② ❧ ✐ ♥ ❣ ❤ ❛ ❚ ② ♣ ❡ ✲ ❆ ❛ ❣ ❡ ♥ ✇ ✐ ❧ ❧ ❜ ❡ ❝ ♦ ♠ ❡ ❡ ① ✐ ♥ ❝ ✳
✷ ✮ limn→∞
E
X Bn
= ∞ ✐ ♠ ♣ ❧ ② ✐ ♥ ❣ ❤ ❛ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ♠ ❛ ② ✢ ♦ ✉ ✐ ❤ ✳
◆ ❖ ❚ ❊ ✿ µBA = 0 ✐ ♠ ♣ ❧ ✐ ❡ ❤ ❛ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ❞ ♦ ❡ ♥ ♦ ❣ ❡ ♥ ❡ ❛ ❡ ❚ ② ♣ ❡ ✲ ❆ ❛ ❣ ❡ ♥ ✳
❈ ♦ ♥ ❞ ✐ ✐ ♦ ♥ ❢ ♦ ✇ ❤ ✐ ❝ ❤ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ❲ ■ ▲ ▲ ❡ ① ✐ ♥ ❝ ❜ ✉ ❚ ② ♣ ❡ ✲ ❆ ❛ ❣ ❡ ♥ ▼ ❆ ❨ ✢ ♦ ✉ ✐ ❤ ❢ ♦ ❛ ♥ ② ✐ ♥ ✐ ✐ ❛ ❧ ♣ ♦ ♣ ✉ ❧ ❛ ✐ ♦
❚ ❤ ✐ ✐ ✐ ♠ ✐ ❧ ❛ ♦ ❤ ❡ ♣ ❡ ✈ ✐ ♦ ✉ ❝ ❛ ❡ ✳ ❚ ❤ ❡ ❡ ✉ ✐ ❡ ❞ ❝ ♦ ♥ ❞ ✐ ✐ ♦ ♥ ✐
✶ ✼
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µAB = 0 and µBB ≤ 1 and µAA > 1 ✭ ✼ ✻ ✮
◆ ❖ ❚ ❊ ✿ µAB = 0 ✐ ♠ ♣ ❧ ✐ ❡ ❤ ❛ ❚ ② ♣ ❡ ✲ ❆ ❛ ❣ ❡ ♥ ❞ ♦ ❡ ♥ ♦ ❣ ❡ ♥ ❡ ❛ ❡ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ✳
✭ ❜ ✮ ■ ♥ ❤ ✐ ♣ ❛ ♦ ❢ ❤ ❡ ✉ ❡ ✐ ♦ ♥ ✇ ❡ ❞ ♦ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ✿
• ❉ ♦ ▼ ♦ ♥ ❡ ❈ ❛ ❧ ♦ ✐ ♠ ✉ ❧ ❛ ✐ ♦ ♥ ♦ ❣ ❡ ♥ ❡ ❛ ❡ ❢ ♦ ✉ ✉ ❛ ❧ ✐ ❛ ✐ ✈ ❡ ❧ ② ❞ ✐ ✛ ❡ ❡ ♥ ❝ ❛ ❡ ✿ ✐ ✮ ❇ ♦ ❤ ❆ ❣ ❡ ♥ ❡ ① ✐ ♥ ❝ ✳ ✐ ✐ ✮ ❇ ♦ ❤ ❛ ❣ ❡ ♥
✢ ♦ ✉ ✐ ❤ ❡ ✳ ✐ ✐ ✐ ✮ ❚ ② ♣ ❡ ✲ ❆ ❡ ① ✐ ♥ ❝ ❛ ♥ ❞ ❚ ② ♣ ❡ ✲ ❇ ✢ ♦ ✉ ✐ ❤ ❡ ✳ ✐ ✈ ✮ ❚ ② ♣ ❡ ✲ ❇ ❡ ① ✐ ♥ ❝ ❛ ♥ ❞ ❚ ② ♣ ❡ ✲ ❆ ✢ ♦ ✉ ✐ ❤ ❡ ✳
• ❈ ♦ ♠ ♠ ❡ ♥ ❤ ♦ ✇ ❤ ❡ ♠ ❡ ❛ ♥ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ♦ ✛ ♣ ✐ ♥ ❣ ♦ ❢ ❡ ❛ ❝ ❤ ② ♣ ❡ ✇ ✐ ❧ ❧ ❛ ✛ ❡ ❝ ✇ ❤ ✐ ❝ ❤ ❝ ❛ ❡ ✇ ✐ ❧ ❧ ❜ ❡ ♦ ❜ ❡ ✈ ❡ ❞ ✳ ❲ ❡ ❤ ❛ ✈ ❡ ❛ ❧ ❡ ❛ ❞ ②
❛ ♥ ✇ ❡ ❡ ❞ ❤ ✐ ✐ ♥ ❛ ✭ ❢ ✮ ♦ ❢ ❤ ❡ ✉ ❡ ✐ ♦ ♥ ✳ ■ ♥ ❞ ❡ ❡ ❞ ✱ ✇ ❡ ✉ ❡ ❤ ❡ ❡ ✉ ❧ ❢ ♦ ♠ ♣ ❛ ✭ ❢ ✮ ♦ ❣ ❡ ♥ ❡ ❛ ❡ ❤ ❡ ❢ ♦ ✉ ❞ ✐ ✛ ❡ ❡ ♥
❝ ❛ ❡ ✳
• ❈ ♦ ♠ ♣ ✉ ❡ ❤ ❡ ❡ ① ✐ ♥ ❝ ✐ ♦ ♥ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ✐ ❡ ❢ ♦ ❤ ❡ ❢ ♦ ✉ ❞ ✐ ✛ ❡ ❡ ♥ ❝ ❛ ❡ ✳ ❚ ❤ ✐ ✇ ❛ ❛ ❦ ❡ ❞ ✐ ♥ ❛ ✭ ❞ ✮ ♦ ❢ ❤ ❡ ✉ ❡ ✐ ♦ ♥ ✳
❲ ❡ ❛ ♥ ❞ ♦ ♠ ❧ ② ❣ ❡ ♥ ❡ ❛ ❡ ❞ ❢ ♦ ✉ ❡ ♦ ❢ p(A) ❛ ♥ ❞ p(B)❡ ❛ ❝ ❤ ❝ ♦ ❡ ♣ ♦ ♥ ❞ ✐ ♥ ❣ ♦ ♦ ♥ ❡ ♦ ❢ ❤ ❡ ❢ ♦ ✉ ❝ ♦ ♥ ❞ ✐ ✐ ♦ ♥ ✇ ❤ ✐ ❝ ❤ ✇ ❡ ❞ ❡ ✐ ✈ ❡ ❞ ✐ ♥
❛ ✭ ❢ ✮ ♦ ❢ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ✳ ❍ ♦ ✇ ❡ ✈ ❡ ✇ ❡ ❞ ✐ ❞ ✐ ♠ ♣ ♦ ❡ ♦ ♠ ❡ ❡ ✐ ❝ ✐ ♦ ♥ ♦ ♥ ❤ ❡ ✉ ❝ ✉ ❡ ♦ ❢ p(A) ❛ ♥ ❞ p(B)✱ ✐ ✳ ❡ ✳ ♦ ♥ ❡ ❚ ② ♣ ❡ ✲ ❆
❛ ❣ ❡ ♥ ❝ ❛ ♥ ♣ ♦ ❞ ✉ ❝ ❡ ❛ ♠ ❛ ① ✐ ♠ ✉ ♠ ♦ ❢ ✷ ❚ ② ♣ ❡ ✲ ❆ ❛ ❣ ❡ ♥ ❛ ♥ ❞ ✶ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ❆ ◆ ❉ ❙ ■ ▼ ■ ▲ ❆ ❘ ▲ ❨ ♦ ♥ ❡ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ❝ ❛ ♥ ♣ ♦ ❞ ✉ ❝ ❡ ❛
♠ ❛ ① ✐ ♠ ✉ ♠ ♦ ❢ ✷ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ❛ ♥ ❞ ✶ ❚ ② ♣ ❡ ✲ ❆ ❛ ❣ ❡ ♥ ✳ ❲ ❡ ❤ ❛ ❞ ♦ ✐ ♠ ♣ ♦ ❡ ❤ ❡ ❡ ❝ ♦ ♥ ❞ ✐ ✐ ♦ ♥ ❜ ❡ ❝ ❛ ✉ ❡ ♦ ❤ ❡ ✇ ✐ ❡ ✐ ✇ ❛ ❜ ❡ ❝ ♦ ♠ ✐ ♥ ❣
✈ ❡ ② ❞ ✐ ✣ ❝ ✉ ❧ ♦ ❛ ♥ ❞ ♦ ♠ ❧ ② ❣ ❡ ♥ ❡ ❛ ❡ p(A) ❛ ♥ ❞ p(B)✇ ❤ ✐ ❝ ❤ ❤ ❛ ✈ ❡ ♦ ♠ ❡ ❞ ❡ ✐ ❡ ❞ ♣ ♦ ♣ ❡ ✐ ❡ ✳ ❉ ✉ ❡ ♦ ❤ ✐ ❡ ✐ ❝ ✐ ♦ ♥ ♦ ♥ ❤ ❡
✉ ❝ ✉ ❡ ♦ ❢ p(A) ❛ ♥ ❞ p(B)✱ ❤ ❡ ✐ ♠ ✉ ❧ ❛ ✐ ♦ ♥ ❡ ✉ ❧ ❛ ❡ ♥ ♦ ✈ ❡ ② ✐ ♥ ❡ ❡ ✐ ♥ ❣ ✳
❚ ❤ ❡ ✐ ♠ ✉ ❧ ❛ ✐ ♦ ♥ ❡ ✉ ❧ ❛ ❡ ❤ ♦ ✇ ♥ ✐ ♥ ❤ ❡ ♥ ❡ ① ♣ ❛ ❣ ❡ ✳ ✳
✶ ✽
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Epoch
0 2 4 6 8 10 12 14 16 18 20
N o . o f T y p e - A a g e n t
0
0.5
1
1.5
2A
B
A & B
Epoch
0 2 4 6 8 10 12 14 16 18 20
N o . o f T y p e - B a g e n t
0
0.5
1
1.5
2A
B
A & B
Epoch
0 2 4 6 8 10 12 14 16 18 20
E x t i n c t i o n P r o b a b i l i t y
0
0.2
0.4
0.6
0.8
1Extinction probability starting with one Type-A agent
Epoch
0 2 4 6 8 10 12 14 16 18 20
E x t i n c t i o n P r o b a b i l i t y
0
0.2
0.4
0.6
0.8
1Extinction probability starting with one Type-B agent
Both Type-A and Type-B agent goes extinct
❋ ✐ ❣ ✉ ❡ ✸ ✿ ❇ ♦ ❤ ❚ ② ♣ ❡ ✲ ❆ ❛ ♥ ❞ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ❣ ♦ ❡ ❡ ① ✐ ♥ ❝ ✳ ❆ ❢ ♦ ❤ ❡ ❧ ❡ ❣ ❡ ♥ ❞ ✬ ❆ ✬ ♠ ❡ ❛ ♥ ❛ ✐ ♥ ❣ ✇ ✐ ❤ ♦ ♥ ❧ ② ♦ ♥ ❡ ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ✱
✬ ❇ ✬ ♠ ❡ ❛ ♥ ❛ ✐ ♥ ❣ ✇ ✐ ❤ ♦ ♥ ❧ ② ♦ ♥ ❡ ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❇ ❛ ♥ ❞ ✬ ❆ ✫ ❇ ✬ ♠ ❡ ❛ ♥ ❛ ✐ ♥ ❣ ✇ ✐ ❤ ♦ ♥ ❡ ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ❛ ♥ ❞ ♦ ♥ ❡ ♦ ❢ ❚ ② ♣ ❡ ✲ ❇ ✳
Epoch
0 2 4 6 8 10 12 14 16 18 20
N o . o f T y p e - A a g e n t
0
100
200
300
400
500A
B
A & B
Epoch0 2 4 6 8 10 12 14 16 18 20
N o . o f T y p e - B a g e n t
0
200
400
600A
B
A & B
Epoch
0 2 4 6 8 10 12 14 16 18 20
E x t i n c t i o n P r o b a b i l i t y
0
0.1
0.2
0.3
0.4Extinction probability starting with one Type-A agent
Epoch
0 5 10 15 20
E x t i n c t i o n P r o b a b i l i t y
0
0.1
0.2
0.3
0.4
0.5Extinction probability starting with one Type-B agent
Both Type-A and Type-B agent flourishes
❋ ✐ ❣ ✉ ❡ ✹ ✿ ❇ ♦ ❤ ❚ ② ♣ ❡ ✲ ❆ ❛ ♥ ❞ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ✢ ♦ ✉ ✐ ❤ ❡ ✳ ❆ ❢ ♦ ❤ ❡ ❧ ❡ ❣ ❡ ♥ ❞ ✬ ❆ ✬ ♠ ❡ ❛ ♥ ❛ ✐ ♥ ❣ ✇ ✐ ❤ ♦ ♥ ❧ ② ♦ ♥ ❡ ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ✱
✬ ❇ ✬ ♠ ❡ ❛ ♥ ❛ ✐ ♥ ❣ ✇ ✐ ❤ ♦ ♥ ❧ ② ♦ ♥ ❡ ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❇ ❛ ♥ ❞ ✬ ❆ ✫ ❇ ✬ ♠ ❡ ❛ ♥ ❛ ✐ ♥ ❣ ✇ ✐ ❤ ♦ ♥ ❡ ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ❛ ♥ ❞ ♦ ♥ ❡ ♦ ❢ ❚ ② ♣ ❡ ✲ ❇ ✳
✶ ✾
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Epoch0 2 4 6 8 10 12 14 16 18 20
N o . o f T y p e - A a g e n t
0
0.2
0.4
0.6
0.8
1A
B
A & B
Epoch0 2 4 6 8 10 12 14 16 18 20
N o . o f T y p e - B a g e n t
0
500
1000
1500A
B
A & B
Epoch
0 2 4 6 8 10 12 14 16 18 20
E x t i n c t i o n P r o b a b i l i t y
0
0.05
0.1
0.15
0.2Extinction probability starting with one Type-A agent
Epoch
0 2 4 6 8 10 12 14 16 18 20
E x t i n c t i o n P r o b a b i l i t y
0
0.05
0.1
0.15Extinction probability starting with one Type-B agent
Type-A extincts and Type-B flourishes
❋ ✐ ❣ ✉ ❡ ✺ ✿ ❚ ② ♣ ❡ ✲ ❆ ❛ ❣ ❡ ♥ ❣ ♦ ❡ ❡ ① ✐ ♥ ❝ ❜ ✉ ❚ ② ♣ ❡ ✲ ❇ ✢ ♦ ✉ ✐ ❤ ❡ ✳ ❆ ❢ ♦ ❤ ❡ ❧ ❡ ❣ ❡ ♥ ❞ ✬ ❆ ✬ ♠ ❡ ❛ ♥ ❛ ✐ ♥ ❣ ✇ ✐ ❤ ♦ ♥ ❧ ② ♦ ♥ ❡ ❛ ❣ ❡ ♥ ♦ ❢
❚ ② ♣ ❡ ✲ ❆ ✱ ✬ ❇ ✬ ♠ ❡ ❛ ♥ ❛ ✐ ♥ ❣ ✇ ✐ ❤ ♦ ♥ ❧ ② ♦ ♥ ❡ ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❇ ❛ ♥ ❞ ✬ ❆ ✫ ❇ ✬ ♠ ❡ ❛ ♥ ❛ ✐ ♥ ❣ ✇ ✐ ❤ ♦ ♥ ❡ ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ❛ ♥ ❞ ♦ ♥ ❡
♦ ❢ ❚ ② ♣ ❡ ✲ ❇ ✳
Epoch
0 2 4 6 8 10 12 14 16 18 20
N o . o f T y p e - A a g e n t
0
500
1000
1500
2000A
B
A & B
Epoch
0 2 4 6 8 10 12 14 16 18 20
N o . o
f T y p e - B a g e n t
0
0.2
0.4
0.6
0.8
1A
B
A & B
Epoch0 2 4 6 8 10 12 14 16 18 20
E x t i n c t i o n P r o b a b i l i t y
0
0.1
0.2
0.3
0.4Extinction probability starting with one Type-A agent
Epoch0 2 4 6 8 10 12 14 16 18 20
E x t i n c t i o n P r o b a b i l i t y
0
0.1
0.2
0.3
0.4Extinction probability starting with one Type-B agent
Type-B extincts and Type-A flourishes
❋ ✐ ❣ ✉ ❡ ✻ ✿ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ❣ ♦ ❡ ❡ ① ✐ ♥ ❝ ❜ ✉ ❚ ② ♣ ❡ ✲ ❆ ✢ ♦ ✉ ✐ ❤ ❡ ✳ ❆ ❢ ♦ ❤ ❡ ❧ ❡ ❣ ❡ ♥ ❞ ✬ ❆ ✬ ♠ ❡ ❛ ♥ ❛ ✐ ♥ ❣ ✇ ✐ ❤ ♦ ♥ ❧ ② ♦ ♥ ❡ ❛ ❣ ❡ ♥ ♦ ❢
❚ ② ♣ ❡ ✲ ❆ ✱ ✬ ❇ ✬ ♠ ❡ ❛ ♥ ❛ ✐ ♥ ❣ ✇ ✐ ❤ ♦ ♥ ❧ ② ♦ ♥ ❡ ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❇ ❛ ♥ ❞ ✬ ❆ ✫ ❇ ✬ ♠ ❡ ❛ ♥ ❛ ✐ ♥ ❣ ✇ ✐ ❤ ♦ ♥ ❡ ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ❛ ♥ ❞ ♦ ♥ ❡
♦ ❢ ❚ ② ♣ ❡ ✲ ❇ ✳
✷ ✵
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✸ ❚ ❤ ❡ ♦ ❡ ✐ ❝ ❛ ❧ ♦ ❜ ❧ ❡ ♠
✸ ✳ ✶ ❆ ♥ ♦ ❤ ❡ ❈ ✐ ❡ ✐ ♦ ♥ ❢ ♦ ❚ ❛ ♥ ✐ ❡ ♥ ❝ ❡
✭ ❛ ✮ ♦ ✈ ✐ ♥ ❣ ❤ ❛ f ∗ ∈ A✱ ♠ ❡ ❧ ❞ ♦ ✇ ♥ ♦ ♣ ♦ ✈ ✐ ♥ ❣ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ✿
♦ ♦ ❢ ❤ ❛ f ∗ (k) = 0 ✿
f ∗ (k) = supf ∈A
f (k) = supf ∈A
0 ∵ f (k)=0
❛
f ∈A
= 0
♦ ♦ ❢ ❤ ❛ 0 ≤ f ∗ ( j) ≤ 1 ; ∀ j ∈ S ✿
f ∗ ( j) = supf ∈A
f ( j) ≤ supf ∈A
1 ∵ f (j)≤1 ; ∀j ❛
f ∈A
= 1
f ∗ ( j) = supf ∈A
f ( j) ≥ supf ∈A
0 ∵ f (j)≥0 ; ∀j
❛
f ∈A
= 0
❍ ❡ ♥ ❝ ❡ 0 ≤ f ∗ ( j) ≤ 1 ; ∀ j ∈ S ✳
♦ ♦ ❢ ❤ ❛ f ∗ ✐ ✉ ❜ ❤ ❛ ♠ ♦ ♥ ✐ ❝ ✇ ✐ ❤ ❡ ♣ ❡ ❝ ♦ P ✿
❈ ♦ ♥ ✐ ❞ ❡ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ❞ ❡ ✜ ♥ ✐ ✐ ♦ ♥
f i ≡ arg supf ∈A
f (i) ✭ ✼ ✼ ✮
◗ ✉ ❛ ❧ ✐ ❛ ✐ ✈ ❡ ❧ ② ♣ ❡ ❛ ❦ ✐ ♥ ❣ ✱ f i ✐ ❛ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ❜ ❡ ❧ ♦ ♥ ❣ ✐ ♥ ❣ ♦ ❤ ❡ ❡ A ✇ ❤ ✐ ❝ ❤ ♠ ❛ ① ✐ ♠ ✐ ③ ❡ f (i) ✳ ■ ♥ ♦ ❤ ❡ ✇ ♦ ❞ f ∗ (i) = f i (i) ✳ ◆ ♦ ✇
❝ ♦ ♥ ✐ ❞ ❡ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣
j∈S P ijf ∗ ( j) ≥
j∈S P ijf i ( j) ✭ ✼ ✽ ✮
≥ f i (i) ✭ ✼ ✾ ✮
= f ∗ (i)
■ ♥ ❡ ✉ ❛ ❧ ✐ ② ✼ ✽ ❡ ✉ ❧ ❢ ♦ ♠ ❤ ❡ ❞ ❡ ✜ ♥ ✐ ✐ ♦ ♥ ♦ ❢ f ∗ ( j) ✱ ✐ ✳ ❡ ✳ f ∗ ( j) = supf ∈A
f ( j) ≥ h ( j) ; ∀h ∈ A✳ ■ ♥ ✐ ♥ ❡ ✉ ❛ ❧ ✐ ② ✼ ✽ ✱ h = f i ✳
■ ♥ ❡ ✉ ❛ ❧ ✐ ② ✼ ✾ ❡ ✉ ❧ ❢ ♦ ♠ ❤ ❡ ❢ ❛ ❝ ❤ ❛ f i ∈ A ❛ ♥ ❞ ❤ ❡ ♥ ❝ ❡ f i ✐ ❛ ✉ ❜ ❤ ❛ ♠ ♦ ♥ ✐ ❝ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ✇ ✐ ❤ ❡ ♣ ❡ ❝ ♦ P ✳ ❚ ❤ ❡ ❛ ❜ ♦ ✈ ❡
♣ ♦ ♦ ❢ ❤ ♦ ✇ ❤ ❛ f ∗ ✐ ✉ ❜ ❤ ❛ ♠ ♦ ♥ ✐ ❝ ✇ ✐ ❤ ❡ ♣ ❡ ❝ ♦ P ✳
✭ ❜ ✮ ■ ♥ ❛ ✭ ❛ ✮ ✇ ❡ ❛ ❧ ❡ ❛ ❞ ② ♣ ♦ ✈ ❡ ❞ ❤ ❛ f ∗ ✐ ✉ ❜ ❤ ❛ ♠ ♦ ♥ ✐ ❝ ✇ ✐ ❤ ❡ ♣ ❡ ❝ ♦ P ✱ ✐ ✳ ❡ ✳j∈S
P ijf ∗ ( j) ≥ f ∗ (i) ; ∀i = k
◆ ♦ ✇ ✇ ❡ ❤ ❛ ✈ ❡ ♦ ♣ ♦ ✈ ❡ ❤ ❛ f ∗ ✐ ❤ ❛ ♠ ♦ ♥ ✐ ❝ ✇ ✐ ❤ ❡ ♣ ❡ ❝ ♦ P ✳ ❚ ❤ ✐ ♠ ❡ ❧ ❞ ♦ ✇ ♥ ♦ ♣ ♦ ✈ ✐ ♥ ❣ ❤ ❛ ❤ ❡ ❡ ❡ ① ✐ ♥ ♦ i ∈ S ❢ ♦
✇ ❤ ✐ ❝ ❤ j∈S
P ijf ∗ ( j) > f ∗ (i) ✭ ✽ ✵ ✮
❛ ♥ ❞ ❤ ❡ ♥ ❝ ❡ ❤ ❡ ♦ ♥ ❧ ② ♣ ♦ ✐ ❜ ✐ ❧ ✐ ② ✐
j∈S
P ijf ∗ ( j) = f ∗ (i) ; ∀i = k ✳ ❲ ❡ ✇ ✐ ❧ ❧ ♣ ♦ ✈ ❡ ❤ ✐ ❜ ② ✉ ✐ ♥ ❣ ❝ ♦ ♥ ❛ ❞ ✐ ❝ ✐ ♦ ♥ ✳ ▲ ❡ ❞ ❡ ✜ ♥ ❡ ❛
❢ ✉ ♥ ❝ ✐ ♦ ♥ g ✉ ❝ ❤ ❤ ❛
g (i) =j∈S
P ijf ∗ ( j) ; ∀i ∈ S ✭ ✽ ✶ ✮
✷ ✶
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❆ ❝ ❝ ♦ ❞ ✐ ♥ ❣ ♦ ✐ ♥ ❡ ✉ ❛ ❧ ✐ ② ✽ ✵ ✱ g (i) > f ∗ (i) ✳ ■ ❢ ✇ ❡ ❝ ❛ ♥ ❤ ♦ ✇ ❤ ❛ g ∈ A✱ ❤ ❡ ♥ ✐ ✇ ✐ ❧ ❧ ❝ ♦ ♥ ❛ ❞ ✐ ❝ ❤ ❡ ❞ ❡ ✜ ♥ ✐ ✐ ♦ ♥ ♦ ❢ f ∗ ✱ ❤ ❡ ❡ ❜ ②
♣ ♦ ✈ ✐ ♥ ❣ ❤ ❛ ✐ ♥ ❡ ✉ ❛ ❧ ✐ ② ✽ ✵ ✐ ♥ ♦ ♣ ♦ ✐ ❜ ❧ ❡ ✳ ◆ ♦ ✇ ❛ ❧ ❧ ✇ ❡ ❤ ❛ ✈ ❡ ♦ ♣ ♦ ✈ ❡ ✐ g ∈ A✳
♦ ♦ ❢ ❤ ❛ g (k) = 0 ✿
g (k) =j∈S
P kjf ∗ ( j) = f ∗ (k) = 0 ∵ f ∗ ∈ A
♦ ♦ ❢ ❤ ❛ 0 ≤ g ( j) ≤ 1 ; ∀ j ∈ S ✿
g ( j) = u∈S
P juf ∗ (u) ≤ u∈S
P ju · 1 ∵ f ∗(u)≤1 ❛
f ∗∈A
= 1 ✭ ❘ ♦ ✇ ✉ ♠ ♦ ❢ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ① ✐ 1)
g ( j) =u∈S
P juf ∗ (u) ≥u∈S
P ju · 0 ∵ f ∗(u)≥0 ❛ f ∗∈A
= 0
❍ ❡ ♥ ❝ ❡ 0 ≤ g ( j) ≤ 1 ; ∀ j ∈ S ✳
♦ ♦ ❢ ❤ ❛ g ✐ ✉ ❜ ❤ ❛ ♠ ♦ ♥ ✐ ❝ ✇ ✐ ❤ ❡ ♣ ❡ ❝ ♦ P ✿
j∈S
P ijg ( j)
=j∈S
P ij
u∈S
P juf ∗ (u)
=j∈S
P ij
u∈S
P juf u (u)
✭ ✽ ✷ ✮
≥j∈S
P ij
u∈S
P juf j (u)
✭ ✽ ✸ ✮
≥ j∈S
P ijf j ( j) ✭ ✽ ✹ ✮
=j∈S
P ijf ∗ ( j) ✭ ✽ ✺ ✮
= g (i)
❊ ✉ ❛ ✐ ♦ ♥ ✽ ✷ ❛ ♥ ❞ ✽ ✺ ✐ ❛ ❞ ✐ ❡ ❝ ❝ ♦ ♥ ❡ ✉ ❡ ♥ ❝ ❡ ♦ ❢ ❤ ❡ ❞ ❡ ✜ ♥ ✐ ✐ ♦ ♥ ♦ ❢ f ✐ ♥ ❡ ✉ ❛ ✐ ♦ ♥ ✼ ✼ ✳ ■ ♥ ❡ ✉ ❛ ❧ ✐ ② ✽ ✸ ❡ ✉ ❧ ❢ ♦ ♠ ❤ ❡ ❢ ❛ ❝ ❤ ❛
f u (u) ≥ f j (u) ; ∀ j ∈ S ✳ ■ ♥ ❡ ✉ ❛ ❧ ✐ ② ✽ ✹ ✐ ✈ ❛ ❧ ✐ ❞ ❜ ❡ ❝ ❛ ✉ ❡ f ∗ ✐ ❛ ✉ ❜ ❤ ❛ ♠ ♦ ♥ ✐ ❝ ✇ ✐ ❤ ❡ ♣ ❡ ❝ ♦ P ✳
❚ ❤ ❡ ❛ ❜ ♦ ✈ ❡ ❤ ❡ ❡ ♣ ♦ ♦ ❢ ❤ ♦ ✇ ❤ ❛ g ∈ A✳ ❚ ❤ ✐ ❝ ♦ ♥ ❝ ❧ ✉ ❞ ❡ ❤ ❡ ♣ ♦ ♦ ❢ ✳
✷ ✷
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11/25/15 10:50 AM C:\Users\sahah_000\Desk...\Markov_Maze.m 1 of 4
clear
clc
load(!ro"a"#l#$%_&ra's#$#o'_Ma$r#.ma$)*
+++++++++++++++++++++++++ Code for 1.1.". : ,&A-& ++++++++++++++++++++++
!_$#lde!*
!_$#lde(:)0*
!_$#lde()1*
!_$#lde*
(:)3*
(:)3*
fo'es(151)*
m_$a#'v(e%e(15))6f*
s$rAvera7e &#me "efore -a$h"er$ reaches cheese 'm2s$r(m_$a(15))3*
d#s8(s$r)*
$em8m_$a(15)*
++++++++++++++++++++++++++ Code for 1.1.". : 9D +++++++++++++++++++++++
+++++++++++++++++++++++++ Code for 1.1.c. : ,&A-& ++++++++++++++++++++++
clearvars ece8$ ! ; $em8
!_$#lde!*
!_$#lde(:)0*
!_$#lde()1*
!_$#lde*
(:)3*
(:)3*
fzeros(151)*
f(<1)1*
f(=1)1*
m_shock#'v(e%e(15))6f*
s$rAvera7e 'm"er of shock "efore reach#'7 cheese 'm2s$r(m_shock(15))3*
d#s8(s$r)*
++++++++++++++++++++++++++ Code for 1.1.c. : 9D +++++++++++++++++++++++
+++++++++++++++++++++++++ Code for 1.1.d. : ,&A-& ++++++++++++++++++++++
clearvars ece8$ ! ; $em8
!_$#lde!*
!_$#lde(:)0*
Appendix A: Markov Maze
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11/25/15 10:50 AM C:\Users\sahah_000\Desk...\Markov_Maze.m 2 of 4
!_$#lde(=:)0*
!_$#lde(>:)0*
!_$#lde()1*
!_$#lde(==)1*
!_$#lde(>>)1*
ro?_dele$e = >3*
col_dele$e = >3*
!_$#lde*
(ro?_dele$e:)3*
(:col_dele$e)3*
clear ro?_dele$e col_dele$e
ro?_dele$e = >3*
col_dele$e1 2 4 5 @ < 10 11 12 1 14 15 1@3*
-!_$#lde*
-(ro?_dele$e:)3*
-(:col_dele$e)3*
U#'v(e%e(1))6-*
s$r!ro"a"#l#$% of reach#'7 cheese ?#$ho$ 7e$$#'7 shocked 'm2s$r(U(11))3*
d#s8(s$r)*
++++++++++++++++++++++++++ Code for 1.1.d. : 9D +++++++++++++++++++++++
+++++++++++++++++++++++++ Code for 1.1.e. : ,&A-& ++++++++++++++++++++++
clearvars ece8$ ! ; $em8
Mzeros(25@25@)*
for #0:1:255
for 0:1:255
s_r_#floor(#/1@)B1*
s_s_##1@6(s_r_#1)B1*
s_r_floor(/1@)B1*
s_s_1@6(s_r_1)B1*
M(#B1B1)!(s_r_#s_r_)6;(s_s_#s_s_)*
e'd
e'd
M_$#ldeM*
for #0:1<:255
M_$#lde(#B1:)zeros(125@)*
M_$#lde(#B1#B1)1*
e'd
M_$#lde*
ro?_dele$e0:1<:255*
col_dele$e0:1<:255*
ro?_dele$ero?_dele$eB1*
col_dele$ecol_dele$eB1*
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11/25/15 10:50 AM C:\Users\sahah_000\Desk...\Markov_Maze.m of 4
(ro?_dele$e:)3*
(:col_dele$e)3*
fo'es(2401)*
m_$a#'v(e%e(240))6f*
s$rAvera7e &#me "efore -a$"er$ a'd ,am mee$ each o$her 'm2s$r(m_$a(22@))3*
d#s8(s$r)*
++++++++++++++++++++++++++ Code for 1.1.e. : 9D +++++++++++++++++++++++
+++++++++++++++++++++++++ Code for 1.1.f. : ,&A-& ++++++++++++++++++++++
clearvars ece8$ ! $em8
8_s$'r%o'es(11@)6#'v(e%e(1@)!Bo'es(1@1@))* + -es'#ck ormla
$a1/8_s$'r%()*
s$rAvera7e &#me "efore -a$"er$ ca' res$ 'm2s$r($em8)B'm2s$r($a)(C1)3*
d#s8(s$r)*
++++++++++++++++++++++++++ Code for 1.1.f. : 9D +++++++++++++++++++++++
+++++++++++++++++++++++++ Code for 1.1.7. : ,&A-& ++++++++++++++++++++++
clearvars ece8$ !
!_$#lde!*
!_$#lde(:)0*
!_$#lde(=:)0*
!_$#lde(>:)0*
!_$#lde()1*
!_$#lde(==)1*
!_$#lde(>>)1*
ro?_dele$e = >3*
col_dele$e = >3*
!_$#lde*
(ro?_dele$e:)3*
(:col_dele$e)3*
clear ro?_dele$e col_dele$e
ro?_dele$e = >3*
col_dele$e1 2 4 5 @ < 10 11 12 1 14 15 1@3*
-!_$#lde*
-(ro?_dele$e:)3*
-(:col_dele$e)3*
U#'v(e%e(1))6-*
8=U(12)*
8>U(1)*
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11/25/15 10:50 AM C:\Users\sahah_000\Desk...\Markov_Maze.m 4 of 4
!_$#lde(=:)!(=:)*
!_$#lde(>:)!(>:)*
clear
!_$#lde*
(:)3*
(:)3*
fzeros(151)*
f(=1)1*
m_shock#'v(e%e(15))6f*
=m_shock(<)*
fzeros(151)*
f(<1)1*
m_shock#'v(e%e(15))6f*
>m_shock(=)*
s$r98ec$ed 'm"er of shock (mod#f#ed) 'm2s$r(8=6(=B1)B8>6(>B1))3*
d#s8(s$r)*
++++++++++++++++++++++++++ Code for 1.1.7. : 9D +++++++++++++++++++++++
clear
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11/25/15 10:50 AM C:\Users\sah...\Email_Sharing_Analysis.m 1 of 2
funcion !mean"mean_a##iional"$ariance"$ariance_a##iional"E%incion&ro'"
E%incion&ro'_a##iional( ) Email_Sharing_Analysis*m"+",-",",
ecor Creaion: S+A3+
mean)4eros*1"+16
mean_a##iional)4eros*1"+16
$ariance)4eros*1"+16
$ariance_a##iional)4eros*1"+16
E%incion&ro')4eros*1"+16
E%incion&ro'_a##iional)4eros*1"+16
ecor Creaion: E7
-niiali4aion for 0h E,och: S+A3+
mean*1)m6
mean_a##iional*1)m6
$ariance*1)06
$ariance_a##iional*1)06
E%incion&ro'*1)06
E%incion&ro'_a##iional*1)06
-niiali4aion for 0h E,och: E7
&reliminary Calculaion: S+A3+
i)numel*,-6
mu_-)sum**0:1:*i819,-6
$ar_-)sum****0:1:*i818mu_-.9**0:1:*i818mu_-.9,-6
o)numel*,6
mu_)sum**0:1:*o819,6
$ar_)sum****0:1:*o818mu_.9**0:1:*o818mu_.9,6
a##)numel*,6
mu_)sum**0:1:*a##819,6
$ar_)sum****0:1:*a##818mu_.9**0:1:*a##818mu_.9,6
A)!mu_9mu_- 06 $ar_9mu_-$ar_-9*mu_;2 *mu_9mu_-;2(6
A_a##)A6
A_a##*2"1)A_a##*2"129mu_9mu_9mu_-6
<_a##)!mu_6 $ar_mu_;2(6
&reliminary Calculaion: E7
al,ha)06
'ea)16
for n)1:1:+
Appendix B: Email Sharing
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11/25/15 10:50 AM C:\Users\sah...\Email_Sharing_Analysis.m 2 of 2
=)!mean*n6mean*n;2$ariance*n(6
=)A9=6
mean*n1)=*16
$ariance*n1)=*28=*1;26
=a##)!mean_a##iional*n6mean_a##iional*n;2$ariance_a##iional*n(6
=a##)A_a##9=a##<_a##6
mean_a##iional*n1)=a##*16
$ariance_a##iional*n1)=a##*28=a##*1;26
'ea)'ea9&ro'_>en_?unc*,"al,ha6
al,ha)&ro'_>en_?unc*,-"&ro'_>en_?unc*,"al,ha6
E%incion&ro'*n1)al,ha;m6
E%incion&ro'_a##iional*n1)*al,ha;m9'ea6
en#
en#
funcion !$al( ) &ro'_>en_?unc*,#f"s
$al)06
)numel*,#f6
for i)0:1:*81
$al)$al,#f*i19*s;i6
en#
en#
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11/25/15 10:51 AM C:\Use...\Branching_TwoWays_MonteCarlo. 1 o! 2
!"nction #$_A%$_B& ' Branching_TwoWays_MonteCarlo(T%)_A%)_B%$_A_0%$_B_0*
+ ,n)"ts:
+ T -- "er o! i"lation te)s (incl"ing the 0th e)och*
+ )_A -- 34 o! o!!s)rings o! Ty)e-A agent
+ )_B -- 34 o! o!!s)rings o! Ty)e-B agent
+ $_A_0 -- 6 o! Ty)e-A agents in the 0th e)och
+ $_B_0 -- 6 o! Ty)e-B agents in the 0th e)och
+ 7"t)"ts:
+ $_A -- 6 o! Ty)e-A agents at i!!erent e)ochs
+ $_B -- 6 o! Ty)e-B agents at i!!erent e)ochs
$_A'8eros(1%T91*
$_B'8eros(1%T91*
$_A(1*'$_A_0 + 6 o! Ty)e-A agents in the 0th e)och
$_B(1*'$_B_0 + 6 o! Ty)e-B agents in the 0th e)och
!or n'1:1:T
+ Contri"tion o! Ty)e-A agent !or the ne;t e)och: TA<T
!or =A'1:1:$_A(n*
#>_AA%>_AB&'<an"?en_24()_A* + Ty)e-A an Ty)e-B agent )ro"ce y =Ath agent
o! Ty)e-A in the c"rrent e)och
$_A(n91*'$_A(n91*9>_AA + U)ation o! Ty)e-A agent !or the ne;t e)och
$_B(n91*'$_B(n91*9>_AB + U)ation o! Ty)e-B agent !or the ne;t e)och
en
+ Contri"tion o! Ty)e-A agent !or the ne;t e)och: @4
+ Contri"tion o! Ty)e-B agent !or the ne;t e)och: TA<T
!or =B'1:1:$_B(n*
#>_BA%>_BB&'<an"?en_24()_B* + Ty)e-A an Ty)e-B agent )ro"ce y =Bth agent
o! Ty)e-B in the c"rrent e)och
$_A(n91*'$_A(n91*9>_BA + U)ation o! Ty)e-A agent !or the ne;t e)och
$_B(n91*'$_B(n91*9>_BB + U)ation o! Ty)e-B agent !or the ne;t e)och
en
+ Contri"tion o! Ty)e-B agent !or the ne;t e)och: @4
en
en
+ 24 <ano "er ?enerator: TA<T
!"nction #col%row&'<an"?en_24()*
+ col is the A agent
+ row is the B agent
_col'n"el()(1%:**
_row'n"el()(:%1**
ppendix C: Monte Carlo Simulation of 2-Way Branching Proce
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11/25/15 10:51 AM C:\Use...\Branching_TwoWays_MonteCarlo. 2 o! 2
te)'ran
acc')(1%1*
i! (te)''1*
col'_col
row'_row
else
col'1
row'1
while(te)'acc*
col'col91
i! (col_col*
col'1
row'row91
en
acc'acc9)(row%col*
en
en
col'col-1
row'row-1
en
+ 24 <ano "er ?enerator: @4
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11/25/15 10:51 AM C:\Users...\Branching_TwoWays_Analysis.m 1 o 1
!nc"ion #$ro%_A&$ro%_B' ( Branching_TwoWays_Analysis)$A&$B&T*
+ ,n$!":
+ $A -- ro%a%ili"y is"ri%!"ion o s$rings o Ty$e-A agen"
+ $B -- ro%a%ili"y is"ri%!"ion o s$rings o Ty$e-B agen"
+ T -- im!la"ion Time
+ !"$!":
+ $ro%_A -- 34"inc"ion ro%a%ili"y s"ar"ing wi"h one Ty$e-A agen"
+ $ro%_B -- 34"inc"ion ro%a%ili"y s"ar"ing wi"h one Ty$e-B agen"
$ro%_A(eros)1&T61*7
$ro%_B(eros)1&T61*7
$ro%_A)1*(07 + 34"inc"ion ro%a%ili"y a" 0"h e$och is 0.
$ro%_B)1*(07 + 34"inc"ion ro%a%ili"y a" 0"h e$och is 0.
or n(1:1:T
$ro%_A)n61*(ro%_8en_9!nc)$A&$ro%_A)n*&$ro%_B)n**7 + ec!rsion 9orm!la
$ro%_B)n61*(ro%_8en_9!nc)$B&$ro%_A)n*&$ro%_B)n**7 + ec!rsion 9orm!la
en;
en;
+ ro%a%ili"y 8enera"ing 9!nc"ion o Two <aria%les: TAT
!nc"ion #=al' ( ro%_8en_9!nc)$;&!&=*
>a(n!mel)$;)1&:**7
>%(n!mel)$;):&1**7
=al(07
or ?(0:1:)>%-1*
or i(0:1:)>a-1*
=al(=al6$;)?61&i61*@)!i*@)=?*7
en;
en;
en;
+ ro%a%ili"y 8enera"ing 9!nc"ion o Two <aria%les: 3>
Appendix D: Extinction Probability of 2-Way Branching Process