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𝑖 𝑙𝑖 𝑒𝑖 𝑁 𝑖 𝑁𝑖 𝑁𝑖 𝑁 = 𝑙𝑖 𝑒𝑖 σ 𝑗 𝑙𝑗 𝑒𝑗 𝑒𝑖 = 1 𝑒𝑖 𝑞𝑖 = 𝑙𝑖 𝑒𝑖 𝑞𝑖
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𝑛 𝑔 = σ𝑗 𝑞𝑗 𝑖 𝑞𝑖 𝑁 𝑝 = 𝑞𝑖 /𝑔 𝑖 Pr 𝑁 = 𝑘 = 𝐵𝑖𝑛𝑜𝑚 𝑘|𝑛, 𝑝 = 𝑛 𝑘 𝑝𝑘 1 − 𝑝 𝑛−𝑘
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𝑛 𝑔 𝜆 = 𝑛𝑝 = 𝑛𝑞𝑖 𝑔 𝑛, 𝑔 lim 𝜆=𝑛𝑝: fix 𝑛,𝑔→∞ 𝑛 𝑘 𝑝𝑘 1 − 𝑝 𝑛−𝑘 = 𝜆𝑘𝑒−𝜆 𝑘! = 𝑃𝑜𝑖𝑠𝑠𝑜𝑛 𝑘|𝜆 𝜆 = 𝑛𝑝 𝑖 𝜆 𝜆 = 1 𝜆 = 2 𝜆 = 3
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𝜆 𝜆 𝜆1 𝜆2
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𝜆 𝑖 𝑗 𝜆𝑖𝑗 𝜑 𝜃 𝜃 𝑃 𝑥|𝜃 𝜃 𝑃 𝑥 = 1 𝑀 𝑗=1 𝑀 𝑃 𝑥|𝜃𝑗 ≃ න𝜑 𝜃 𝑃 𝑥|𝜃 𝑑𝜃
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𝐺𝑎𝑚𝑚𝑎 𝜆|𝜇, 𝜙 𝑃𝑜𝑖𝑠𝑠𝑜𝑛 𝑥|𝜆 න 0 ∞ 𝐺𝑎𝑚𝑚𝑎 𝜆|𝜙, 𝜇 𝜙 𝑃𝑜𝑖𝑠𝑠𝑜𝑛 𝑥|𝜆 𝑑𝜆 = Γ 𝑥 + 𝜙 Γ 𝑥 + 1 Γ 𝜙 𝜙 𝜇 + 𝜙 𝜙 𝜇 𝜇 + 𝜙 𝑥 = 𝑁𝑒𝑔𝐵𝑖𝑛𝑜𝑚 𝑥|𝜇, 𝜙 𝜇, 𝜙 𝑉 𝑥 = 𝜇 + 𝜇2 𝜙 > 𝜇 𝜙 = ∞
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𝜙 𝜎2 = 𝜇 𝜎2 = 𝜇 + 𝜇2 𝜙
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𝑒𝑖 𝑞𝑖 𝑥𝑖 ′ = 𝑥𝑖 𝑙𝑖 𝐸 𝑥′ = 𝐸 𝑥 𝑙𝑖 = 𝜇 𝑙𝑖 = 𝜇, 𝑉 𝑥′ = 𝑉 𝑥 𝑙𝑖 2 = 𝜇 𝑙𝑖 2 + 𝜇2 𝑙𝑖 2𝜙 = 𝜇 𝑙𝑖 + 𝜇2 𝜙 ≠ 𝜇 + 𝜇2 𝜙
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log2 𝑦 = 𝑋𝑓𝑢𝑙𝑙 𝛽 log2 𝑦 = 𝑋𝑟𝑒𝑑𝑢𝑐𝑒𝑑 𝛽 𝑃 𝑌|𝑋𝑓𝑢𝑙𝑙 𝑃 𝑌|𝑋𝑟𝑒𝑑𝑢𝑐𝑒𝑑
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𝑃 𝑘 = 𝑛 𝑛 − 1 ⋯ 𝑛 − 𝑘 𝑘! 𝑝𝑘 1 − 𝑝 𝑛−𝑘 𝜆 = 𝑛𝑝 𝑛 → ∞ 𝑝 → 0 𝑃 𝑘 ≈ 𝑛𝑘 𝑘! 𝑝𝑘 1 − 𝑝 𝑛 = 𝜆𝑘 𝑘! 1 − 𝜆 𝑛 𝑛 𝑒 1 − 𝜆 𝑛 𝑛 → 𝑒−𝜆 𝑃 𝑘 ≈ 𝜆𝑘 𝑘! 𝑒−𝜆
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න 0 ∞ 𝐺𝑎𝑚𝑚𝑎 𝜆|𝜙, 𝜇 𝜙 𝑃𝑜𝑖𝑠𝑠𝑜𝑛 𝑥|𝜆 𝑑𝜆 = න 0 ∞ 𝜆𝜙−1𝑒− 𝜆𝜇 𝜙 𝜙 𝜇 𝜙 Γ 𝜙 𝜆𝑥 𝑥! 𝑒−𝜆𝑑𝜆 = 𝜇 𝜙 𝜙 Γ 𝑥 + 1 Γ 𝜙 න 0 ∞ 𝜆𝜙+𝑥−1𝑒−𝜆 𝜇+𝜙 𝜙 𝑑𝜆 = 𝜙 𝜇 + 𝜙 −𝜙 𝜇 𝜇 + 𝜙 𝜙 Γ 𝑥 + 1 Γ 𝜙 𝜙 𝜇 + 𝜙 𝜙+𝑥 Γ 𝜙 + 𝑥 = Γ 𝜙 + 𝑥 Γ 𝑥 + 1 Γ 𝜙 𝜙 𝜇 + 𝜙 𝑥 𝜇 𝜇 + 𝜙 𝜙 = Γ 𝑥 + 𝜙 Γ 𝑥 + 1 Γ 𝜙 𝜙 𝜇 + 𝜙 𝜙 𝜇 𝜇 + 𝜙 𝑥
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