對於參數為 $$$n = 3$$$ 與 $$$p = 0.2$$$ 的幾何分佈，求 $$$P{\left(X = 3 \right)}$$$

計算參數為 $$$n = 3$$$ 和 $$$p = 0.2 = \frac{1}{5}$$$ 的幾何分布各種數值（不將成功那次試驗計入）。

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。

標準差：$$$\sigma = \sqrt{\frac{1 - p}{p^{2}}} = \sqrt{\frac{1 - \frac{1}{5}}{\left(\frac{1}{5}\right)^{2}}} = 2 \sqrt{5}\approx 4.472135954999579$$$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