User:dgtljunglist

${\mbox{E}}\left(X\right)\;=\;\int _{0}^{\infty }{x\;f\left(x\right)\;dx}=\;\int _{0}^{\infty }{x\;{\frac {1}{\beta }}\;e^{\frac {x}{\beta }}dx}=\;\left[-{\frac {x+\beta }{e^{\frac {x}{\beta }}}}\right]_{0}^{\infty }$

$=\lim _{x\to \infty }-{\frac {x+\beta }{e^{\frac {x}{\beta }}}}+{\frac {0+\beta }{e^{\frac {0}{\beta }}}}=\;-\lim _{x\to \infty }{\frac {\beta }{{\frac {1}{\beta }}e^{\frac {x}{\beta }}}}+\beta =\;\beta$