例 2.1.5

由题设可知，$X$ 的可能取值为 $0, 1, 2, 3$。又记 $A_i$ = "汽车在第 $i$ 个路口遇到红灯"，$i = 1, 2, 3$。因为 $A_1, A_2, A_3$ 相互独立，且

$$P(A_i) = P(\overline{A_i}) = \frac{1}{2}, \quad i = 1, 2, 3.$$

所以得

$$P(X = 0) = P(A_1) = \frac{1}{2},$$ $$P(X = 1) = P(\overline{A_1}A_2) = P(\overline{A_1})P(A_2) = \frac{1}{4},$$ $$P(X = 2) = P(\overline{A_1}\overline{A_2}A_3) = P(\overline{A_1})P(\overline{A_2})P(A_3) = \frac{1}{8},$$ $$P(X = 3) = P(\overline{A_1}\overline{A_2}\overline{A_3}) = P(\overline{A_1})P(\overline{A_2})P(\overline{A_3}) = \frac{1}{8},$$

故 $X$ 的分布列如下：

$X$	0	1	2	3
$P$	$\frac{1}{2}$	$\frac{1}{4}$	$\frac{1}{8}$	$\frac{1}{8}$