Question

Question:Suppose that the joint p.d.f. of X and Y is as follows:

\(f\left( {x,y} \right) = \left\{ \begin{array}{l}24xy for x \ge 0,y \ge 0, and x + y \le 1,\\0 otherwise\end{array} \right.\).

Are X and Y independent?

Short answer

X and Y are not independent.

Step-by-step solution

Compute the marginal density of X

The marginal density of\(X,f\left( x \right)\)is

\(\begin{array}{c}X,f\left( x \right) = \int\limits_0^1 {24xydy} \\ = 12x.......(1)\end{array}\)

Compute the marginal density of Y

The marginal density of\(Y,f\left( y \right)\)

\(\begin{array}{c}Y,f\left( y \right) = \int\limits_0^1 {24xydx} \\ = 12y..........(2)\end{array}\)

From equations (1) and (2) we can say that

\(f\left( x \right) \times f\left( y \right) = 144xy\)

\( \ne f(x,y)\)

Hence we can conclude that X and Y are not independent