数学试题讲解

设随机变量

X_{1}, X_{2}, \dots, X_{n}

独立同分布, 且

X_{1}

的 4 阶矩存在, 记

E (X_{1}^{k}) = μ_{k} (k = 1, 2, 3, 4)

,则由切比雪夫不等式, 对任意由

ε > 0

有

P {| \frac{1}{n} \sum_{i = 1}^{n} X_{i}^{2} - μ_{2} | \geq ε} \leq

\frac{μ_{4} - μ_{2}^{2}}{n ε^{2}}

\frac{μ_{4} - μ_{2}^{2}}{\sqrt{n} ε^{2}}

\frac{μ_{2} - μ_{1}^{2}}{n ε^{2}}

\frac{μ_{2} - μ_{1}^{2}}{\sqrt{n} ε^{2}}