Example

Example
設 $X_1,\ldots,X_n$ 為獨立且有相同分布的隨機變數, 其期望值為 $\mu$, 變異數為 $\sigma ^2$, 且令 $ \overline{X} = \displaystyle\sum_{i=1}^n X_i/n $為樣本均數. $X_i-\overline{X},i=1,\ldots,n $,稱為 離差 (deviation), 因為它們是每一個別資料和樣本均數的差. 令 S2 表示這些離差的平方和. 也就是說,
$S^2 = \displaystyle\sum_{i=1}^n (X_i - \overline{X} )^2 $
隨機變數 S2/(n-1) 稱為 樣本變異數 (sample variance). 求 (a) $ Var(\overline{X})$, 和 (b) E[S2 /(n-1)].
Solution:
(a)
$\begin{array}{rcl} Var(\overline{X}) &=&\left (\displaystyle\frac{1}{n}\right )... ...ox{ by independence } \\ \\ & = & \displaystyle\frac{\sigma ^2}{n} \end{array}$
(b)
$\begin{array}{rcl} S^2&=&\displaystyle\sum_{i=1}^n (X_i - \mu + \mu - \overline... ...=&\displaystyle\sum_{i=1}^n (X_i - \mu)^2 - n(\overline{X} -\mu )^2 \end{array}$
將上式兩邊取期望值得
$\begin{array}{rcl} E[S^2]&=&\displaystyle\sum_{i=1}^n E [(X_i - \mu)^2] - nE[(\... ... \\ &=& n \sigma ^2 - nVar(\overline{X}) \\ \\ &=& (n-1)\sigma ^2 \end{array}$
其中最後一個等式利用 (a) 的結果. 將兩邊除以 n-1, 得樣本變異數的期望值就是分布變異數 $\sigma ^2\qquad\rule[0.02em]{1.0mm}{1.5mm}$.

Example (Variance of a Binomial Random Variable)
Compute the variance of a binomial random variable X with parameters n and p.
Solution:
Since such a random variable represents the number of successes in nindependent trials when each trial has a common probability p of being a success, we may write $X=X_1+\cdots+X_n$ where the Xi are independent Bernoulli random variables such that
$X_i=\left\{ \begin{array}{ll} 1&\mbox{ if the }i\mbox{th trial is a success} \\ \\ 0&\mbox{ otherwise } \end{array}\right .$
Hence, we have $Var(X)=Var(X_1)+\cdots+Var(X_n)$. But
$\begin{array}{rcl} Var(X_i)&=&E[X_i^2]-(E[X_i])^2 \\ \\ &=&E[X_i]-(E[X_i])^2 \qquad \mbox{since} X_i^2=X_i \\ \\ &=&p-p^2 \end{array}$
and thus $Var(X)=np(1-p)\qquad\rule[0.02em]{1.0mm}{1.5mm}$