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Weak Law of Large Number

  • Expectation of the Sample Mean: E[Xn]\mathbb{E}[\overline{X^n}]
  • Variance of the Sample Mean: V[Xn]\mathbb{V}[\overline{X^n}]

Sample Mean θ^n\hat{\theta}_n is converging to Population Mean θ\theta: limnE[(θ^nθ)2]=0\lim\limits_{n \to \infty} \mathbb{E}[(\hat{\theta}_n - \theta)^2] = 0

limnV[θ^n]+(E[θ^n]θ)2)=0\lim\limits_{n \to \infty} \mathbb{V}[\hat{\theta}_n] + (\mathbb{E}[\hat{\theta}_n] - \theta)^2) = 0

limnσx2n+(μxμx)2=0\lim\limits_{n\to\infty} {\sigma_x^2 \over n} + (\mu_x - \mu_x)^2 = 0

θ^nθ\hat{\theta}_n \to \theta

ϵ>0\forall \epsilon > 0, limnP(θ^nθ)>ϵ)=0\lim\limits_{n \to \infty} P(|\hat{\theta}_n - \theta)| > \epsilon) = 0

limnP(xnμx>ϵ)limnσx2nϵ2\lim\limits_{n \to \infty} P(|\overline{x_n} - \mu_x| > \epsilon) \leq \lim\limits_{n \to \infty} {\sigma_x^2 \over {n \epsilon^2}}

MI

x0,cR+,E[X]<x \geq 0, c \in \mathbb{R}^+, \mathbb{E}[X] < \infty

P(X)C)E[X]CP(X) \geq C) \leq {\mathbb{E}[X] \over C}

CI

σx2<\sigma_x^2 < \infty

P(xμx>ϵ)σx2ϵ2P(|x - \mu_x| > \epsilon) \leq {\sigma_x^2 \over \epsilon^2}