Let \(F(x)\) be the distribution function of a nonnegative random variable \(X\) with finite mean value \(\mu\).

(1) Show that the function \(F_{s}(x)\) defined by

\[F_{S}(x)=\frac{1}{\mu} \int_{0}^{x}(1-F(t)) d t\]

is the distribution function of a nonnegative random variable \(X_{s}\).

(2) Prove: If \(X\) is exponentially distributed with parameter \(\lambda=1 / \mu\), then so is \(X_{S}\) and vice versa.

(3) Determine the failure rate \(\lambda_{s}(x)\) of \(X_{s}\).