91Ó°ÊÓ

X and Y both are random variables.

\(Var\left( X \right) = Var\left( Y \right)\)

Let,

\(\begin{align}U &= X + Y\\V &= X - Y\end{align}\)

The expectation of\(X + Y\)and\(X - Y\)is given below

\(\begin{align}E\left( {UV} \right) &= E\left( {\left( {X + Y} \right)\left( {X - Y} \right)} \right)\\ &= E\left( {{X^2} - {Y^2}} \right)\\ &= E\left( {{X^2}} \right) - E\left( {{Y^2}} \right)\end{align}\)

Expectation of \(X + Y\) and expectation of \(X - Y\) as follows

\(\begin{align}E\left( U \right)E\left( V \right) &= E\left( {X + Y} \right)E\left( {X - Y} \right)\\ &= \left( {{\mu _x} + {\mu _y}} \right)\left( {{\mu _x} - {\mu _y}} \right)\\ &= {\mu _x}^2 - {\mu _y}^2\end{align}\)

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

\(\begin{align}Cov\left( {U,V} \right) &= E\left( {UV} \right) - E\left( U \right)E\left( V \right)\\ &= \left( {E\left( {{X^2}} \right) - {\mu _x}^2} \right) - \left( {E\left( {{Y^2}} \right) - {\mu _Y}^2} \right)\\ &= Var\left( X \right) - Var\left( Y \right)\\ &= 0\end{align}\)

Referring Definition 4.6.2 for the equation.

The Correlation of \(X + Y\),\(X - Y\)is

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

Therefore,\(U\)and\(V\)are uncorrelated .i.e,

\(X + Y\)and\(X - Y\)are uncorrelated.

Hence proved.