Central Limit Theorem
Normal Approximation of Binomial Distribution
Other Sampling Distributions
Let \(X_1, X_2, \ldots, X_n\) be identical and independent distributed random variables with \(E(X_i)=\mu\) and \(Var(X_i) = \sigma²\). We define
\[ Y_n = \sqrt n \left(\frac{\bar X-\mu}{\sigma}\right) \mathrm{ where }\ \bar X = \frac{1}{n}\sum^n_{i=1}X_i. \]
Then, the distribution of the function \(Y_n\) converges to a standard normal distribution function as \(n\rightarrow \infty\).
Suppose \(X\sim Bin(n,p)\), furthermore, let \(\bar X = X/n\). If \(n\) is large enough, then
\[ \bar X \overset{\circ}{\sim}N\left\{p,p(1-p)/n\right\}\]\[ \]
\(\chi^2\)-distribution
t-distribution
F-distribution
Let \(Z_1, Z_2,\ldots,Z_n \overset{iid}{\sim}N(0,1)\),
\[ \sum_{i=1}^nZ_i^2\sim\chi^2_n. \]
\(X_1, X_2,\ldots,X_n\) form a random sample from a normal distribution, then \(\bar X\) and \(S^2\) are independent of each other.
Let \(X_1, X_2,\ldots,X_n \overset{iid}{\sim}N(\mu,\sigma^2)\), \(S^2 = \frac{1}{n-1}\sum^n_{i=1}(X_i-\bar X)^2\), and \(\bar X \perp S^2\); therefore:
\[ \frac{(n-1)S^2}{\sigma^2} \sim \chi^2_{n-1}. \]
Let \(Z\sim N(0,1)\), \(W\sim \chi^2_\nu\), \(Z\perp W\); therefore:
\[ T=\frac{Z}{\sqrt{W/\nu}} \sim t_\nu \]
Let \(W_1\sim\chi^2_{\nu_1}\) \(W_2\sim\chi^2_{\nu_2}\), and \(W_1\perp W_2\); therefore:
\[ F = \frac{W_1/\nu_1}{W_2/\nu_2}\sim F_{\nu_1,\nu_2} \]