Probability Density Function
\[f_X(x)=\begin{cases} \lambda e^{-\lambda x},&\quad x\geq0\\ 0,&\quad x<0 \end{cases}\]Expectation
\[\mathbb{E}[X]=\frac{1}{\lambda}\] Proof:
$$\begin{align} \mathbb{E}[X]&=\int_{0}^{\infty}x\lambda e^{-\lambda x}\mathrm{d}x\\ &= \end{align}$$
Variance
\[\mathrm{var}(X)=\frac{1}{\lambda^2}\] Proof:
$$\begin{align} \mathbb{E}[X]&=\int_{0}^{\infty}x^2\lambda e^{-\lambda x}\mathrm{d}x\\ &= \end{align}$$