91Ó°ÊÓ

Two random variables X and Y,their joint distribution follows the uniform distribution over a rectangle with sides parallel to the coordinate axis in the x-y plane.

We know that the formula for the correlation coefficient is,\(Cor\left( {X,Y} \right) = \frac{{Cov\left( {X,Y} \right)}}{{{\sigma _X}{\sigma _Y}}}\)where \({\sigma _X}\;and\;{\sigma _Y}\) are the standard deviations.

From this, one can conclude that the correlation is not affected by changing the distribution of X and Y in the x-y plane. So, one can consider that at the origin the distribution rectangle has the center.

So, we illustrated it in the plot,

Therefore, by the plot, we consider that the mean of both variables will be 0.

\(E\left( X \right) = E\left( Y \right) = 0\).

Let us consider a negative value for every positive value of X in the 1^st or 4^th quadrants of the same magnitude in the 2^nd or 3^rd quadrants. As the joint distribution follows uniform, so, there will be the same probability of occurrence.

Similarly there is negative value for every positive value of Y in the 1^st or 2^ndquadrants of the same magnitude in the 3^rd or 4^th quadrants.

So, one get\(E\left( {XY} \right) = 0\).

Now, it is know that,

\(\begin{aligned}{}Cov\left( {X,Y} \right) = E\left( {XY} \right) - E\left( X \right)E\left( Y \right)\\ = 0\end{aligned}\).

Therefore, the correlation between X and Y is

\(\begin{aligned}{}\rho \left( {X,Y} \right) = \frac{{Cov\left( {X,Y} \right)}}{{{\sigma _X}{\sigma _Y}}}\\ = 0\end{aligned}\)

Hence correlation between X and Y is 0.