Exponential Continuous Probability Distribution

Exponential RV $X$ takes positive values according to PDF:

PDF

$$f(x,\lambda )=\{\begin{array}{ll}\lambda e^{-\lambda x}& x\ge 0\\ 0& x<0\end{array}$$

For some $\lambda >0$

CDF

$$F(x,\lambda )=\{\begin{array}{ll}1-e^{-\lambda x}& x\ge 0\\ 0& x<0\end{array}$$

Expectation

$E(X)=\frac{1}{\lambda }$

Variance

$$\frac{1}{{\lambda }^2}$$

MGF

$$\frac{\lambda }{\lambda -t},t<\lambda$$