Question

Find the mean and standard deviation of each of the following probability distribution :

xi :	0	1	2	3	4	5
pi :	\[\frac{1}{6}\]	\[\frac{5}{18}\]	\[\frac{2}{9}\]	\[\frac{1}{6}\]	\[\frac{1}{9}\]	\[\frac{1}{18}\]

Answer 1

xi	pi	pixi	pixi2
0	\[\frac{1}{6}\]	0	0
1	\[\frac{5}{18}\]	\[\frac{5}{18}\]	\[\frac{5}{18}\]
2	\[\frac{2}{9}\]	\[\frac{4}{9}\]	\[\frac{8}{9}\]
3	\[\frac{1}{6}\]	\[\frac{1}{2}\]	\[\frac{3}{2}\]
4	\[\frac{1}{9}\]	\[\frac{4}{9}\]	\[\frac{16}{9}\]
5	\[\frac{1}{18}\]	\[\frac{5}{18}\]	\[\frac{25}{18}\]
		`∑`pixi = \[\frac{35}{18}\]	`∑`pixi2= \[\frac{35}{6}\]

\[\text{ Mean}  = \sum p_i x_i = \frac{35}{18}\]
\[\text{ Variance } = \sum p_i {x_i}^2 - \left( \text{ Mean } \right)^2 \]
\[ = \frac{35}{6} - \left( \frac{35}{18} \right)^2 \]
\[ = \frac{35}{6} - \frac{1225}{324}\]
\[ = \frac{1890 - 1225}{324}\]
\[ = \frac{665}{324}\]
\[\text{ Step Deviation } = \sqrt{\text{ Variance } }\]
\[ = \sqrt{\frac{665}{324}}\]
\[ = \frac{\sqrt{665}}{18}\]