A continuous random variable \(X\) has the density function
\[
f(x)=
\begin{cases}
e^{-x}, & x>0,
0, & \text{elsewhere},
\end{cases}
\]
then the expected value of
\[
g(x)=e^{\frac{2x}{3}}
\]
is
Show Hint
For an exponential random variable with
\[
f(x)=e^{-x},\;x>0,
\]
\[
\boxed{
E[g(X)]
=
\int_{0}^{\infty}g(x)e^{-x}\,dx.
}
\]