91Ó°ÊÓ

X has a uniform distribution on interval\(\left( { - 2,2} \right)\).

\(Y = {X^6}\)

Let\(X\) it be a random variable, and given that\(Y = {X^6}\).

The p.d.f of X is symmetric with respect to 0.

So,\(E\left( X \right) = 0\)

\(E\left( {{X^k}} \right) = 0;\)k=odd positive integer

\(E\left( {XY} \right) = E\left( {{X^7}} \right) = 0\)

So, \(E\left( {XY} \right) = 0\) and \(E\left( X \right)E\left( Y \right) = 0\)

The covariance of \(X\)and \(Y\)is

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

The correlation between \(X\)and \(Y\)is:

\(\begin{align}\rho \left( {X,Y} \right) &= \frac{{Cov\left( {X,Y} \right)}}{{\rho \left( X \right)\rho \left( Y \right)}}\\ &= 0\end{align}\)

Hence, X and Y are uncorrelated.

Hence, proved.