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1.3.2 乘法公式

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Probability Statistics

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第一章随机事件与概率

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定理 1.3.1 (乘法公式)

定理 1.3.1 (乘法公式)

Mar 08, 20251 min read

定理 1.3.1 (乘法公式)

设 $(Ω, F, P)$ 为概率空间, $A_{i} \in F, i = 1, 2, \dots, n$ 且 $P (A_{1} A_{2} \dots A_{n - 1}) > 0$ , 则
$P (A_{1} A_{2} \dots A_{n}) = P (A_{1}) P (A_{2} ∣ A_{1}) P (A_{3} ∣ A_{1} A_{2}) \dots P (A_{n} ∣ A_{1} A_{2} \dots A_{n - 1}) . (1.3.3)$

证明

反复应用 (1.3.2), 有

P (A_{1} A_{2} \dots A_{n}) = P (A_{1} A_{2} \dots A_{n - 1}) P (A_{n} ∣ A_{1} A_{2} \dots A_{n - 1}) = P (A_{1} A_{2} \dots A_{n - 2}) P (A_{n - 1} ∣ A_{1} A_{2} \dots A_{n - 2}) P (A_{n} ∣ A_{1} A_{2} \dots A_{n - 1}) = \dots = P (A_{1} A_{2}) P (A_{3} ∣ A_{1} A_{2}) \dots P (A_{n} ∣ A_{1} A_{2} \dots A_{n - 1}) = P (A_{1}) P (A_{2} ∣ A_{1}) P (A_{3} ∣ A_{1} A_{2}) \dots P (A_{n} ∣ A_{1} A_{2} \dots A_{n - 1}) .

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