Let \(X\) be a random variable with

\[
f_{X}(x)=\left\{\begin{array}{cc}
\frac{1}{\sigma} t^{3} e^{-t} & t \geq 0 \\
0 & \text { elsewhere }
\end{array}ight\}
\]

(a) Determine the characteristic function \(C_{X}(\omega)\). (b) Using \(C_{X}(\omega)\), validate that \(f_{X}(x)\) is a proper \(p d f\). (c) Use \(C_{X}(\omega)\) to determine the first two moments of \(X\). (d) Calculate the variance of \(X\).