User:Dgtljunglist


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


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