Slide 6
Slide 6 text
分岐過程
• Galton-Watson processの定式化が⼀番有名
を⾮負の整数値をとるiidな確率変数: (ただし )
46
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Zt+1 =
Zt
X
i=1
Xt,i
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Xt,i
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
P(Xt,i = m) = pm
Zt
は世代tの感染者数
Xt,i
は世代tの個体iが感染させた数
N世代後に感染者数はどうなるか?が⽬的
• その定義よりZはマルコフ連鎖(もっと⾔うと普通は状態0を吸
収状態とする吸収的マルコフ連鎖を仮定)
• 世代tの感染者数Zt
の期待値は、ある個体がうつす
感染数の期待値から
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
p0 > 0, p0 + p1 < 1
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
E[Zt] = E[Xt,i]t