Weak Law of Large Number Expectation of the Sample Mean: E[Xn‾]\mathbb{E}[\overline{X^n}]E[Xn] Variance of the Sample Mean: V[Xn‾]\mathbb{V}[\overline{X^n}]V[Xn] Sample Mean θ^n\hat{\theta}_nθ^n is converging to Population Mean θ\thetaθ: limn→∞E[(θ^n−θ)2]=0\lim\limits_{n \to \infty} \mathbb{E}[(\hat{\theta}_n - \theta)^2] = 0n→∞limE[(θ^n−θ)2]=0 limn→∞V[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) = 0n→∞limV[θ^n]+(E[θ^n]−θ)2)=0 limn→∞σx2n+(μx−μx)2=0\lim\limits_{n\to\infty} {\sigma_x^2 \over n} + (\mu_x - \mu_x)^2 = 0n→∞limnσx2+(μx−μx)2=0 θ^n→θ\hat{\theta}_n \to \thetaθ^n→θ ∀ϵ>0\forall \epsilon > 0∀ϵ>0, limn→∞P(∣θ^n−θ)∣>ϵ)=0\lim\limits_{n \to \infty} P(|\hat{\theta}_n - \theta)| > \epsilon) = 0n→∞limP(∣θ^n−θ)∣>ϵ)=0 limn→∞P(∣xn‾−μx∣>ϵ)≤limn→∞σx2nϵ2\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}}n→∞limP(∣xn−μx∣>ϵ)≤n→∞limnϵ2σx2 MI x≥0,c∈R+,E[X]<∞x \geq 0, c \in \mathbb{R}^+, \mathbb{E}[X] < \inftyx≥0,c∈R+,E[X]<∞ P(X)≥C)≤E[X]CP(X) \geq C) \leq {\mathbb{E}[X] \over C}P(X)≥C)≤CE[X] CI σx2<∞\sigma_x^2 < \inftyσx2<∞ P(∣x−μx∣>ϵ)≤σx2ϵ2P(|x - \mu_x| > \epsilon) \leq {\sigma_x^2 \over \epsilon^2}P(∣x−μx∣>ϵ)≤ϵ2σx2GroupthinkPrevious PageWebAssemblyNext Page