对于参数为$$$n = 3$$$和$$$p = 0.2$$$的几何分布，求$$$P{\left(X = 3 \right)}$$$

计算以 $$$n = 3$$$ 和 $$$p = 0.2 = \frac{1}{5}$$$ 为参数的几何分布的各项数值（不计入成功的那次试验）。

平均值：$$$\mu = \frac{1 - p}{p} = \frac{1 - \frac{1}{5}}{\frac{1}{5}} = 4$$$A。

方差：$$$\sigma^{2} = \frac{1 - p}{p^{2}} = \frac{1 - \frac{1}{5}}{\left(\frac{1}{5}\right)^{2}} = 20$$$A。

标准差: $$$\sigma = \sqrt{\frac{1 - p}{p^{2}}} = \sqrt{\frac{1 - \frac{1}{5}}{\left(\frac{1}{5}\right)^{2}}} = 2 \sqrt{5}\approx 4.472135954999579.$$$A

$$$P{\left(X = 3 \right)} = 0.1024$$$A

$$$P{\left(X \lt 3 \right)} = 0.488$$$A

$$$P{\left(X \leq 3 \right)} = 0.5904$$$A

$$$P{\left(X \gt 3 \right)} = 0.4096$$$A

$$$P{\left(X \geq 3 \right)} = 0.512$$$A