對於參數為 $$$n = 5$$$ 與 $$$p = 0.22$$$ 的幾何分佈，求 $$$P{\left(X = 5 \right)}$$$

計算給定 $$$n = 5$$$ 與 $$$p = 0.22 = \frac{11}{50}$$$ 的幾何分布各種數值（包含成功試驗）。

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。

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