Let $X \in \text{Exponential}(\theta)$ be a D214: Exponential random positive real number.

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} $\square$