| $\text{X}$ | $0$ | $1$ | $2$ |
| $\text{P}(\text{X}) $ | $\frac{20}{42}$ | $\frac{20}{42}$ | $\frac{2}{42}$ |
| $\text{x}_\text{i}$ | $\text{p}_\text{i}$ | $\text{p}_\text{i}\text{x}_\text{i}$ | $\text{p}_\text{i}\text{X}_\text{i}^2$ |
| $0$ | $\frac{20}{42}$ | $0$ | $0$ |
| $1$ | $\frac{20}{42}$ | $\frac{20}{42}$ | $\frac{20}{42}$ |
| $2$ | $\frac{2}{42}$ | $\frac{4}{42}$ | $\frac{8}{42}$ |
|
|
| $\sum\text{p}_\text{i}\text{x}_\text{i}=\frac{4}{7}$ | $\sum\text{p}_\text{i}\text{x}_\text{i}^2=\frac{2}{3}$ |
Mean
$=\sum\text{p}_\text{i}\text{x}_\text{i}=\frac{4}{7}$Variance
$=\sum\text{p}_\text{i}\text{x}_\text{i}^2-\big(\sum\text{p}_\text{i}\text{x}_\text{i}\big)^2$$=\frac{2}{3}-\Big(\frac{4}{7}\Big)^2=\frac{50}{147}$
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$\frac{\text{dy}}{\text{dx}}-\frac{2\text{xy}}{1+\text{x}^2}=\text{x}^2+2$