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

Guenuegoomyns 2021-11-09 Answered
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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Expert Answer

Xyle1991
Answered 2021-11-10 Author has 1858 answers
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
\(\displaystyle{V}{a}{r}{\left({Y}\right)}={E}{\left({Y}^{{{2}}}\right)}-{E}{\left({Y}\right)}^{{{2}}}={\frac{{{1}}}{{{2}}}}-{\frac{{{4}}}{{{9}}}}={\frac{{{1}}}{{{18}}}}\)
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