\begin{align*} \text{Mean} = \mu & = \int_0^{\infty} x \cdot \lambda e^{-\lambda x} dx\\ & = \left [ -(x+\frac{1}{\lambda}) e^{-\lambda \cdot x} \right ] \big |_0^{\infty} \\ & = \frac{1}{\lambda}. \end{align*}