A string of length \(l\) is stretched, under a constant tension \(F\), between two fixed points \(A\) and \(B\). Show that the mean square (fluctuational) displacement \(y(x)\) at point \(P\), distant \(x\) from \(A\), is given by

\[
\overline{\{y(x)\}^{2}}=\frac{k T}{F l} x(l-x) .
\]

Further show that, for \(x_{2} \geq x_{1}\),

\[
\overline{y\left(x_{1}\right) y\left(x_{2}\right)}=\frac{k T}{F l} x_{1}\left(l-x_{2}\right) .
\]