sampling

counting §

	order matters	order doesn’t matter
w/o replacement	$\frac{n!}{(n-k)!}=P_k^n$	$\frac{n!}{(n-k)!k!}=\left(\genfrac{}{}{0ex}{}{n}{k}\right)$
with replacement	$n^k$	$\left(\genfrac{}{}{0ex}{}{n+k-1}{k}\right)$

sampling distribution §

• probability distribution of a statistic from many samples of the same size
• shows how the statistic varies across different samples

sample mean §

• if the variance is known, then $\bar{X}\sim N(\mu ,\frac{{\sigma }^2}{n})$ and $Z=\frac{\bar{X}-\mu }{\sigma /\sqrt{n}}\sim N(0,1)$ by the CLT.
• o/w, $T=\frac{\bar{X}-\mu }{s/\sqrt{n}}\sim t(n-1)$

sample variance §

$\frac{(n-1)s^2}{{\sigma }^2}\sim {\chi }^2(n-1)$

least squares estimators §

for i = 0 or 1, we have

$$\frac{{\hat{\beta }}_i-{\beta }_i}{SE(\widehat{{\beta }_0})}\sim t(n-2)$$

resampling methods §

• bootstrapping
• jackknife
• cross validation