Example

Example (The Sample Mean)
Let $X_1,\ldots,X_n$ be independent and identically distributed random variables having distribution function F and expected value $\mu$. Such a sequence of random variables is said to constitute a sample from the distribution F. The quantity $\overline{X}$, defined by
$\overline{X}=\displaystyle\sum_{i=1}^n\frac{X_i}{n}$
is called the sample mean. Compute $E[\overline{X}]$.
Solution:
$\begin{array}{rcl} E[\overline{X}]&=&E\left [\displaystyle\sum_{i=1}^n\frac{X_i... ...1}^nE[X_i] \\ \\ &=&\mu \qquad\qquad\mbox{ since } E[X_i]\equiv\mu \end{array}$
That is, the expected value of the sample mean is $\mu$, the mean of the distribution. When the distribution mean $\mu$ is unknown, the sample mean is often used in statistics to estimate it.

Example (Boole's Inequality)
令 $A_1,\ldots,A_n $ 為 n 個事件且定義指標變數 $X_i (i=1,2,\ldots,n)$ 如下:
$ X_i = \left \{ \begin{array}{ll} 1 & \qquad \mbox{ if } A_i \mbox{ occurs } \\ \\ 0 & \qquad \mbox{ otherwise } \end{array} \right .$
令 $ X= \displaystyle\sum_{i=1}^n X_i $. 故 X 表示事件 Ai 發生的個數. 最後, 令
$ Y = \left \{ \begin{array}{ll} 1 & \qquad \mbox{ if } X \geq 1 \\ \\ 0 & \qquad \mbox{ otherwise } \end{array} \right .$
也就是說, 當至少有一事件 Ai 發生時, Y 等於 1, 否則 Y 為 0. 因 $ X \geq Y $     故 $ E[X] \geq E[Y] $但是因為 $ E[X] = \displaystyle\sum_{i=1}^n E[X_i] = \displaystyle\sum_{i=1}^n P(A_i) $且
$\begin{array}{rcl} E[Y]&=&P\{\mbox{at least one of the } A_i \mbox{ occur}\} \\ \\ &=&P\left (\displaystyle\bigcup_{i=1}^n A_i \right ) \end{array}$
所以我們得到 Boole 不等式, 即 $P \left ( \displaystyle\bigcup_{i=1}^n A_i\right ) \leq\sum_{i=1}^n P(A_i)\qquad\rule[0.02em]{1.0mm}{1.5mm}$