Find the mean and variance for the following probability density functions $f(x) = \left\{ \begin{array}{l l} \alpha e^{-\alpha x} ,& \quad \text{if$x$$>$$0$}\\ 0 ,& \quad \text{otherwise} \end{array} \right.$

Toolbox:
• Let X be a continuous random variable with probability density function f(x). Then the mathematical expectation of X is defined as $E(X)=\int_{-\infty}^\infty x f(x)dx$
• $E(\phi (X))=\int_{\infty}^{\infty}\phi(x) f(x)dx$
• Var$(X)=E(X^2)-[E(X)]^2$
• $E(c)=c$
• $E(aX\pm b)=aE (X)\pm b$
Step 1:
$E(X)=\int_{-\infty}^\infty x f(x) dx$
$\qquad=\int_0^\infty x\alpha e^{-\alpha x}dx$