The probability distribution of a random variable \(X\) is given by
\(\begin{array}{|c|c|c|c|}
\hline \mathrm{X}=x & 0 & 1 & 2\mathrm{P}(\mathrm{X}=x) & \frac{1}{5} & \frac{2}{5} & \frac{2}{5}\\ \hline\end{array}\)
then the variance of \(\mathrm{X}\) is

\(\begin{array}{|c|c|c|c|}
\hline \mathrm{x}_{1} & \mathrm{p}\left(\mathrm{x}_{\mathrm{i}}\right) & \mathrm{p}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}} & \mathrm{p}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}}^{2} \\
\hline 0 & \frac{1}{5} & 0 & 0 \\
\hline 1 & \frac{2}{5} & \frac{2}{5} & \frac{2}{5} \\
\hline 2 & \frac{2}{5} & \frac{4}{5} & \frac{8}{5} \\
\hline
\end{array}\)
\(\therefore \Sigma \mathrm{p}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}} =\frac{6}{5} \quad \text { and } \quad \Sigma \mathrm{p}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}}^{2}=\frac{10}{5}=2 \)
\( \text { Variance } =\Sigma \mathrm{p}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}}^{2}-\left(\Sigma \mathrm{p}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}}\right)^{2} \)
\( =2-\left(\frac{6}{5}\right)^{2} \quad=2-\frac{36}{25}=\frac{14}{25}\)