对于参数为 $$$n = 100$$$ 和 $$$p = 0.2$$$ 的二项分布，求 $$$P{\left(X = 20 \right)}$$$

根据 $$$n = 100$$$、$$$p = 0.2 = \frac{1}{5}$$$ 和 $$$x = 20$$$ 计算二项分布的各项数值。

平均值：$$$\mu = n p = \left(100\right)\cdot \left(\frac{1}{5}\right) = 20$$$A。

方差：$$$\sigma^{2} = n p \left(1 - p\right) = \left(100\right)\cdot \left(\frac{1}{5}\right)\cdot \left(1 - \frac{1}{5}\right) = 16$$$A。

标准差: $$$\sigma = \sqrt{n p \left(1 - p\right)} = \sqrt{\left(100\right)\cdot \left(\frac{1}{5}\right)\cdot \left(1 - \frac{1}{5}\right)} = 4$$$A.

$$$P{\left(X = 20 \right)}\approx 0.099300214808825$$$A

$$$P{\left(X \lt 20 \right)}\approx 0.460161370064573$$$A

$$$P{\left(X \leq 20 \right)}\approx 0.559461584873398$$$A

$$$P{\left(X \gt 20 \right)}\approx 0.440538415126602$$$A

$$$P{\left(X \geq 20 \right)}\approx 0.539838629935427$$$A