Measures of spread
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For the probability density function:
\(f(x) = \left\{ {\begin{array}{*{20}{c}}{\dfrac{2}{{13}}({x^2} + 4),\,\,0 \le x \le 1}\\{0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,otherwise}\end{array}} \right.\)
Find the standard deviation
\begin{aligned}
\mu &=\int_{0}^{1} xf(x) d x \\
&=\int_{0}^{1} x\times \frac{2}{13}\left(x^{2}+4\right) d x \\
&=\frac{2}{13} \int_{0}^{1} x^{3}+4 x d x\\
&=\frac{2}{13}\left[\frac{x^{4}}{4}+2 x^{2}\right]_{0}^{1} \\
&=\frac{2}{13}\left[\frac{1}{4}+2-0\right]\\
& =\frac{9}{26}
\end{aligned}
\begin{aligned}
{\sigma ^2} &= \int\limits_0^1 {{x^2}f(x)\,dx - {\mu ^2}} \\ &= \int\limits_0^1 {{x^2}\frac{2}{{13}}({x^2} + 4)dx} - {\mu ^2}\\ &= \frac{2}{{13}}\int\limits_0^1 {{x^4} + 4{x^2}\,dx} - {\mu ^2}\\
&= \frac{2}{{13}}\left[ {\frac{{{x^5}}}{5} + \frac{{4{x^3}}}{3}} \right]_0^1 - {\mu ^2}\\ &= \frac{2}{{13}}\left[ {\frac{1}{5} + \frac{4}{3} - 0} \right] - {\left( {\frac{9}{{26}}} \right)^2}= 0.1161\\\therefore \,\sigma &= 0.3407
\end{aligned}
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