# 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.
Are X and Y independent?

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Lauren Fuller
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.