Mean \(\displaystyle{\left[{E}{\left({x}\right)}\right]}\) of \(\displaystyle{x}\)

The formula for mean is given by,

\(\displaystyle{E}{\left({x}\right)}=\int_{{x}}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}\)

\(\displaystyle={\int_{{4}}^{{0}}}{x}\cdot{0.125}{x}{\left.{d}{x}\right.}\)

\(\displaystyle={\int_{{4}}^{{0}}}{x}^{{2}}\cdot{0.125}{x}{\left.{d}{x}\right.}\)

\(\displaystyle={0.125}={\int_{{4}}^{{0}}}{x}^{{2}}{\left.{d}{x}\right.}\)

\(\displaystyle={0.125}{\left[\frac{{{x}^{{3}}}}{{{3}}}\right]}^{{4}}_{0}\) {since \(\displaystyle\int{x}^{{n}}{\left.{d}{x}\right.}=\frac{{{x}^{{{n}+{1}}}}}{{{n}+{1}}}\)}

\(\displaystyle={0.125}{\left[\frac{{{4}^{{3}}}}{{{3}}}-{0}\right]}\)

\(\displaystyle={0.125}{\left(\frac{{{64}}}{{3}}\right)}\)

\(\displaystyle{E}{\left({x}\right)}=\frac{{8}}{{3}}\) or \(\displaystyle{2.667}\)

The formula for mean is given by,

\(\displaystyle{E}{\left({x}\right)}=\int_{{x}}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}\)

\(\displaystyle={\int_{{4}}^{{0}}}{x}\cdot{0.125}{x}{\left.{d}{x}\right.}\)

\(\displaystyle={\int_{{4}}^{{0}}}{x}^{{2}}\cdot{0.125}{x}{\left.{d}{x}\right.}\)

\(\displaystyle={0.125}={\int_{{4}}^{{0}}}{x}^{{2}}{\left.{d}{x}\right.}\)

\(\displaystyle={0.125}{\left[\frac{{{x}^{{3}}}}{{{3}}}\right]}^{{4}}_{0}\) {since \(\displaystyle\int{x}^{{n}}{\left.{d}{x}\right.}=\frac{{{x}^{{{n}+{1}}}}}{{{n}+{1}}}\)}

\(\displaystyle={0.125}{\left[\frac{{{4}^{{3}}}}{{{3}}}-{0}\right]}\)

\(\displaystyle={0.125}{\left(\frac{{{64}}}{{3}}\right)}\)

\(\displaystyle{E}{\left({x}\right)}=\frac{{8}}{{3}}\) or \(\displaystyle{2.667}\)