对于参数为$$$n = 5$$$和$$$p = 0.22$$$的几何分布，求$$$P{\left(X = 5 \right)}$$$

使用$$$n = 5$$$和$$$p = 0.22 = \frac{11}{50}$$$计算几何分布的各项数值（计入成功试次）。

平均值：$$$\mu = \frac{1}{p} = \frac{1}{\frac{11}{50}} = \frac{50}{11}\approx 4.545454545454545$$$A。

方差：$$$\sigma^{2} = \frac{1 - p}{p^{2}} = \frac{1 - \frac{11}{50}}{\left(\frac{11}{50}\right)^{2}} = \frac{1950}{121}\approx 16.115702479338843$$$A。

标准差: $$$\sigma = \sqrt{\frac{1 - p}{p^{2}}} = \sqrt{\frac{1 - \frac{11}{50}}{\left(\frac{11}{50}\right)^{2}}} = \frac{5 \sqrt{78}}{11}\approx 4.014436757421749.$$$A

$$$P{\left(X = 5 \right)} = 0.0814331232$$$A

$$$P{\left(X \lt 5 \right)} = 0.62984944$$$A

$$$P{\left(X \leq 5 \right)} = 0.7112825632$$$A

$$$P{\left(X \gt 5 \right)} = 0.2887174368$$$A

$$$P{\left(X \geq 5 \right)} = 0.37015056$$$A