As each point inside the circle can\({x^2} + {y^2} \le 1,\,\,f\left( {x,y} \right)\)can be factored.
This factorisation cannot be satisfied at every point outside this circle.
This shows X and Y are not independent.
As the joint pdf of X and Y is
\(f\left( {x,y} \right) = \left\{ \begin{array}{l}k{x^2}{y^2}\,\,for\,{x^2} + {y^2} \le 1\\0\,\,\,\,\,\,\,\,\,\,\,\,\,otherwise\end{array} \right.\)
The Marginal pdf of\(X,f\left( x \right)\)
\(\begin{array}{c}X,f\left( x \right) = \int\limits_{ - 1}^1 {K{x^2}{y^2}dy} \\ = 2k{x^2}\frac{{{{\left( {1 - {x^2}} \right)}^{\frac{3}{2}}}}}{3}\end{array}\)
Therefore,
\({f_1}\left( x \right) = \left\{ \begin{array}{l}2k{x^2}\frac{{{{\left( {1 - {x^2}} \right)}^{\frac{3}{2}}}}}{3}\,\,\,\,\,if - 1 \le x \le 1\\0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,otherwise\end{array} \right.\)
Similarly the marginal density of\(Y,f\left( y \right)\)
\(\begin{array}{c}Y,f\left( y \right) = \int\limits_{ - 1}^1 {k{x^2}{y^2}dx} \\ = 2k{y^2}\frac{{{{\left( {1 - {y^2}} \right)}^{\frac{3}{2}}}}}{3}\end{array}\)
Therefore,
\({f_1}\left( y \right) = \left\{ \begin{array}{l}2k{y^2}\frac{{{{\left( {1 - {y^2}} \right)}^{\frac{3}{2}}}}}{3}\,\,\,\,\,\,\,\,\, - 1 \le y \le 1\\0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,otherwise\end{array} \right.\)
Hence we can say that X and Y are not independent and have the same marginal pdf’s
