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Question
Find the mean and standard deviation of each of the following probability distribution :
| xi : | 0 | 1 | 2 | 3 | 4 | 5 |
| pi : |
\[\frac{1}{6}\]
|
\[\frac{5}{18}\]
|
\[\frac{2}{9}\]
|
\[\frac{1}{6}\]
|
\[\frac{1}{9}\]
|
\[\frac{1}{18}\]
|
Solution
| 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=
\[\frac{35}{6}\]
|
\[\text{ Mean} = \sum p_i x_i = \frac{35}{18}\]
\[\text{ Variance } = \sum p_i {x_i}^2 - \left( \text{ Mean } \right)^2 \]
\[ = \frac{35}{6} - \left( \frac{35}{18} \right)^2 \]
\[ = \frac{35}{6} - \frac{1225}{324}\]
\[ = \frac{1890 - 1225}{324}\]
\[ = \frac{665}{324}\]
\[\text{ Step Deviation } = \sqrt{\text{ Variance } }\]
\[ = \sqrt{\frac{665}{324}}\]
\[ = \frac{\sqrt{665}}{18}\]
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