# The random variables X and Y have joint density function f(x,y)=12xy(1-x) 0<

The random variables X and Y have joint density function
f(x,y)=12xy(1-x) 0 and equal to 0 otherwise.
Find Var(Y).

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Xyle1991
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
$$\displaystyle{V}{a}{r}{\left({Y}\right)}={E}{\left({Y}^{{{2}}}\right)}-{E}{\left({Y}\right)}^{{{2}}}={\frac{{{1}}}{{{2}}}}-{\frac{{{4}}}{{{9}}}}={\frac{{{1}}}{{{18}}}}$$