In order to calculate the variance, let's find the second moment.

\(\displaystyle{E}{\left({Y}^{{{2}}}\right)}={\int_{{{0}}}^{{{1}}}}{y}^{{{2}}}\cdot{2}{y}{\left.{d}{y}\right.}={\frac{{{1}}}{{{2}}}}{y}^{{{4}}}{{\mid}_{{{0}}}^{{{1}}}}={\frac{{{1}}}{{{2}}}}\)

thus

\(\displaystyle{V}{a}{r}{\left({Y}\right)}={E}{\left({Y}^{{{2}}}\right)}-{E}{\left({Y}\right)}^{{{2}}}={\frac{{{1}}}{{{2}}}}-{\frac{{{4}}}{{{9}}}}={\frac{{{1}}}{{{18}}}}\)

\(\displaystyle{E}{\left({Y}^{{{2}}}\right)}={\int_{{{0}}}^{{{1}}}}{y}^{{{2}}}\cdot{2}{y}{\left.{d}{y}\right.}={\frac{{{1}}}{{{2}}}}{y}^{{{4}}}{{\mid}_{{{0}}}^{{{1}}}}={\frac{{{1}}}{{{2}}}}\)

thus

\(\displaystyle{V}{a}{r}{\left({Y}\right)}={E}{\left({Y}^{{{2}}}\right)}-{E}{\left({Y}\right)}^{{{2}}}={\frac{{{1}}}{{{2}}}}-{\frac{{{4}}}{{{9}}}}={\frac{{{1}}}{{{18}}}}\)