91Ó°ÊÓ

The joint p.d.f of X and Y is

\(f\left( {x,y} \right) = \left\{ \begin{array}{l}\frac{1}{3}if\,1 < x < 3\,and\,0 < y < 1\\\frac{1}{6}if\,1 < x < 3\,and\,0 < y < 1\,\\0\,\,otherwise.\\\,\end{array} \right.\,\,\).

The region is not a rectangle. It is a product set that has the form\(\left\{ {\left( {x,y} \right):x \in A,y \in B} \right\}\)for two sets, A and B.

Fig.1

The marginal p.d.f. of X is

\(\begin{array}{l}{f_1}\left( x \right) = \int\limits_0^1 {f\left( {x,y} \right)dy} \\{f_1}\left( x \right) = \left\{ \begin{array}{l}\frac{1}{3}\,if\,1 < x < 3\\\frac{1}{6}\,if\,6 < x < 8\end{array} \right.\end{array}\)

The marginal p.d.f. of Y is

\(\begin{array}{c}{f_2}\left( y \right) = \int\limits_1^3 {\left( {\frac{1}{3}} \right)dx + } \int\limits_6^8 {\left( {\frac{1}{6}} \right)dx} \\ = \frac{2}{3} + \frac{1}{3}\\ = 1\end{array}\)

Therefore, \({f_2}\left( y \right) = \left\{ \begin{array}{l}1\,\,for\,0 < y < 1\\0\,\,otherwise\end{array} \right.\).

The product of the two marginal pdfs is

\({f_1}\left( x \right){f_2}\left( y \right) = \left\{ \begin{array}{l}\frac{1}{3}\,\,if\,1 < x < 3\,and\,0 < y < 1\\\frac{1}{6}\,\,if\,6 < x < 8\,and\,0 < y < 1\\0\,\,\,\,\,otherwise\end{array} \right.\).

Therefore, we can say that\(f\left( {x,y} \right) = {f_1}\left( x \right){f_2}\left( y \right)\).

Hence, the two random variables are independent.