例 2.2.1 (B(10,0,9))

设 $X\sim B\left(10,0.9\right)$, 试求

• $P\left(X=8\right)$,
• $P\left(X\le 8\right)$,
• $P\left(3\le X\le 9\right)$.

解

我们借助 $\mathrm{R}$ 软件来计算.

$$\begin{array}{rl}P\left(X=8\right)& =\left(\genfrac{}{}{0ex}{}{10}{8}\right)0.9^8\cdot 0.1^2\\ & =\mathrm{dbinom}(8,10,0.9)\\ & =\mathrm{0.1937102.}\end{array}$$ $$\begin{array}{rl}P\left(X\le 8\right)& =\sum _{k\le 8}\left(\genfrac{}{}{0ex}{}{10}{k}\right)0.9^k{\left(1-0.9\right)}^{10-k}\\ & =\text{pbinom}(8,10,0.9)\\ & =\mathrm{0.2639011.}\end{array}$$ $$\begin{array}{rl}P\left(3\le X\le 9\right)& =\sum _{3\le k\le 9}\left(\genfrac{}{}{0ex}{}{10}{k}\right)0.9^k{\left(1-0.9\right)}^{10-k}\\ & =\mathrm{pbinom}(9,10,0.9)-\mathrm{pbinom}(2,10,0.9)\\ & =\mathrm{0.6513124.}\end{array}$$