对于参数为 $$$n = 12$$$ 和 $$$p = 0.1$$$ 的二项分布，求 $$$P{\left(X = 8 \right)}$$$

根据 $$$n = 12$$$、$$$p = 0.1 = \frac{1}{10}$$$ 和 $$$x = 8$$$ 计算二项分布的各项数值。

平均值：$$$\mu = n p = \left(12\right)\cdot \left(\frac{1}{10}\right) = \frac{6}{5} = 1.2$$$A。

方差：$$$\sigma^{2} = n p \left(1 - p\right) = \left(12\right)\cdot \left(\frac{1}{10}\right)\cdot \left(1 - \frac{1}{10}\right) = \frac{27}{25} = 1.08$$$A。

标准差: $$$\sigma = \sqrt{n p \left(1 - p\right)} = \sqrt{\left(12\right)\cdot \left(\frac{1}{10}\right)\cdot \left(1 - \frac{1}{10}\right)} = \frac{3 \sqrt{3}}{5}\approx 1.039230484541326.$$$A

$$$P{\left(X = 8 \right)} = 0.000003247695$$$A

$$$P{\left(X \lt 8 \right)} = 0.99999658647$$$A

$$$P{\left(X \leq 8 \right)} = 0.999999834165$$$A

$$$P{\left(X \gt 8 \right)} = 1.65835 \cdot 10^{-7}$$$A

$$$P{\left(X \geq 8 \right)} = 0.00000341353$$$A