Probability Notes 5 极限定理

5 极限定理(Limit Theorem)

5.1 马尔可夫不等式(Markov Inequality)

  • 设随机变量X只可取非负值,则:
    P(Xge{a})lefrac{mathbb{E}[X]}{a}, for all~a data-recalc-dims=0" />

5.2 契比雪夫不等式(Chebyshev Inequality)

  • 设随机变量X期望为mu,方差为sigma^{2},则:
    P(|X-mu|ge{c})lefrac{{sigma}^{2}}{c^2}, for all~c data-recalc-dims=0" />

5.3 弱大数定律(Weak Law of Large Numbers, WLLN)

  • {X}_{1},{X}_{2},dots,{X}_{n}是独立同步分布(independent identically distributed, i.i.d.)的随机变量,共同的期望为mu,则:
    对于任意epsilon data-recalc-dims=0" />有:
    P(|{M}_{n}-mu|ge{epsilon})=P(|frac{{X}_{1}+{X}_{2}+dots+{X}_{n}}{n}-mu|ge{epsilon})rightarrow 0, ~~~~text{as}~nrightarrowinfty

5.4 概率收敛(Convergence in Probability)

  • {Y}_{1},{Y}_{2}dots{Y}_{n}是随机变量的一个数列(sequence),且a为常数。若对于任意epsilon data-recalc-dims=0" />均有:
    lim _{nrightarrowinfty}{P(|{Y}_{n}-a|ge{epsilon})} =0
    则称数列{Y}_{n}依概率收敛于a

5.5 中央极限定理(Central Limit Theroem)

  • {X}_{1},{X}_{2},dots,{X}_{n}是独立同步分布的随机变量,共同的期望为mu,方差为{sigma}^{2},定义标准值{Z}_{n}为:
    {Z}_{n}=frac{{X}_{1}+{X}_{2}+dots+{X}_{n}-nmu}{sigmasqrt{n}}
  • {Z}_{n}的累积分布函数收敛于标准正态累积分布函数:
    lim_{nrightarrow{infty}}P({Z}_{n}le{z})=Phi{(z)},~~for everyz
    Phi{(z)}=frac{1}{sqrt{2pi}}int_{-infty}^{z}{e^{-frac{x^2}{2}}dx}

5.6 德莫佛-拉普拉斯二项分布近似公式(De Moivre-Laplace Approximation to the Binomial)

  • {S}_{n}是二项随机变量,其参数为npn较大且k,l为非负整数时有:
    P(kle{S}_{n}le{l})approxPhi(frac{l+frac{1}{2}-np}{sqrt{np(1-p)}})-Phi(frac{k-frac{1}{2}-np}{sqrt{np(1-p)}})

5.7 强大数定律(Strong Law of Large Numbers)

  • {X}_{1},{X}_{2},dots,{X}_{n}是独立同步分布的随机变量,共同的期望为mu,则:
    P(lim_{nrightarrowinfty}{frac{{X}_{1}+{X}_{2}+dots+{X}_{n}}{n}}=mu)=1

5.8 依概率1收敛

  • {Y}_{1},{Y}_{2}dots{Y}_{n}是随机变量的一个数列(sequence),且c为常数。若:
    P(lim _{nrightarrowinfty}{{Y}_{n}=c})=1
    则称数列{Y}_{n}依概率1收敛于c

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