对于参数为 $$$n = 10$$$ 和 $$$p = 0.6$$$ 的二项分布，求 $$$P{\left(X = 4 \right)}$$$

根据 $$$n = 10$$$、$$$p = 0.6 = \frac{3}{5}$$$ 和 $$$x = 4$$$ 计算二项分布的各项数值。

平均值：$$$\mu = n p = \left(10\right)\cdot \left(\frac{3}{5}\right) = 6$$$A。

方差：$$$\sigma^{2} = n p \left(1 - p\right) = \left(10\right)\cdot \left(\frac{3}{5}\right)\cdot \left(1 - \frac{3}{5}\right) = \frac{12}{5} = 2.4$$$A。

标准差: $$$\sigma = \sqrt{n p \left(1 - p\right)} = \sqrt{\left(10\right)\cdot \left(\frac{3}{5}\right)\cdot \left(1 - \frac{3}{5}\right)} = \frac{2 \sqrt{15}}{5}\approx 1.549193338482967.$$$A

$$$P{\left(X = 4 \right)} = 0.111476736$$$A

$$$P{\left(X \lt 4 \right)} = 0.0547618816$$$A

$$$P{\left(X \leq 4 \right)} = 0.1662386176$$$A

$$$P{\left(X \gt 4 \right)} = 0.8337613824$$$A

$$$P{\left(X \geq 4 \right)} = 0.9452381184$$$A