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小明6:30 ~ 7:30，班车7:00 ~ 8:00 抵达，设 6:30 为 0，则小明在 0 ~ 60，班车在 30 ~ 90 到达。两者到达时间为独立均匀随机变量：

小明抵达时间为随机变量 $X$，概率密度函数为 $f_X(x) = \frac{1}{60-0} = \frac{1}{60}$ 班车抵达时间为随机变量 $Y$，概率密度函数为 $f_Y(y) = \frac{1}{90-30} = \frac{1}{60}$

小明在班车抵达则能搭上班车的概率为

$$ \begin{aligned} P\{X\lt Y\} &= \iint_{x\lt y} f_{X,Y}(x,y)\,dy\,dx \\ &= \int_{0}^{30}\int_{30}^{90} f_X(x)f_Y(y)\,dy\,dx + \int_{30}^{60}\int_{x}^{90} f_X(x)f_Y(y)\,dy\,dx \\ &= \int_{0}^{30}\int_{30}^{90} \left( \frac{1}{60} \right) ^2\,dy\,dx + \int_{30}^{60}\int_{x}^{90} \left( \frac{1}{60} \right) ^2\,dy\,dx \\ &= \left( \frac{1}{60} \right) ^2 \left( \int_{0}^{30}\int_{30}^{90} \,dy\,dx + \int_{30}^{60}\int_{x}^{90} \,dy\,dx \right) \\ &= \left( \frac{1}{60} \right) ^2 \left(1800 + \left( 90x - \frac{x^2}{2}\right) \Big|_{30}^{60} \right) \\ &= \frac{3150}{3600} \\ &= \frac{7}{8} \end{aligned} $$