Let $T \in \text{Exponential}(1)$ be a
D4000: Standard exponential random positive real number. By definition, we have
\begin{equation}
X
\overset{d}{=} \frac{1}{\theta} T
\end{equation}
Result
R5327: Expectation of a standard exponential random positive real number shows that $\mathbb{E} T = 1$, so by homogeneity of expectation, we have
\begin{equation}
\mathbb{E} X
= \mathbb{E} \frac{1}{\theta} T
= \frac{1}{\theta}
\end{equation}
$\square$
