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

crapthach24 2021-11-10 Answered
The random variables X and Y have joint density function
f(x,y)=12xy(1-x) 0 and equal to 0 otherwise.
Are X and Y independent?

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Expert Answer

Lauren Fuller
Answered 2021-11-11 Author has 1245 answers
Observe that the joint PDF of (X,Y) can be written as
f(x,y)=12xy(1-x)=6x(1-x)*2y
for \(\displaystyle{x}\in{\left({0},{1}\right)}\ {\quad\text{and}\quad}\ {y}\in{\left({0},{1}\right)}\), otherwise it is equal to zero. Since we have that \(\displaystyle{\int_{{{0}}}^{{{1}}}}{6}{x}{\left({1}-{x}\right)}{\left.{d}{x}\right.}={1}\ {\quad\text{and}\quad}\ {\int_{{{0}}}^{{{1}}}}{2}{y}{\left.{d}{y}\right.}={1}\), these two function are in fact marginal functions of X and Y. Hence we have that
\(\displaystyle{{f}_{{{X}}}{\left({x}\right)}}={6}{x}{\left({1}-{x}\right)}\)
\(\displaystyle{{f}_{{{Y}}}{\left({y}\right)}}={2}{y}\)
and since the joint PDF factorizes, these random variables are independent.
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