Find $$$P{\left(X = 6 \right)}$$$ for geometric distribution with $$$n = 6$$$ and $$$p = 0.25$$$

Calculate the various values for the geometric distribution with $$$n = 6$$$ and $$$p = 0.25 = \frac{1}{4}$$$ (include a success trial).

Mean: $$$\mu = \frac{1}{p} = \frac{1}{\frac{1}{4}} = 4$$$A.

Variance: $$$\sigma^{2} = \frac{1 - p}{p^{2}} = \frac{1 - \frac{1}{4}}{\left(\frac{1}{4}\right)^{2}} = 12$$$A.

Standard deviation: $$$\sigma = \sqrt{\frac{1 - p}{p^{2}}} = \sqrt{\frac{1 - \frac{1}{4}}{\left(\frac{1}{4}\right)^{2}}} = 2 \sqrt{3}\approx 3.464101615137755.$$$A

$$$P{\left(X = 6 \right)} = 0.059326171875$$$A

$$$P{\left(X \lt 6 \right)} = 0.7626953125$$$A

$$$P{\left(X \leq 6 \right)} = 0.822021484375$$$A

$$$P{\left(X \gt 6 \right)} = 0.177978515625$$$A

$$$P{\left(X \geq 6 \right)} = 0.2373046875$$$A