Expectation of Exponential Distribution/Proof 2

Theorem

Let $X$ be a continuous random variable of the exponential distribution with parameter $\beta$ for some $\beta \in \R_{> 0}$

Then the expectation of $X$ is given by:

$\expect X = \beta$

$\map {M_X} t = \dfrac 1 {1 - \beta t}$

$\expect X = \map {M_X'} 0$

We have:

$\ds \map {M_X'} t$

$=$

$\ds \map {\frac \d {\d t} } {\frac 1 {1 - \beta t} }$

$\ds $

$=$

$\ds \frac {-\beta} {-1} \frac 1 {\paren {1 - \beta t}^2}$

$\ds $

$=$

$\ds \frac \beta {\paren {1 - \beta t}^2}$

Setting $t = 0$ gives:

$\ds \expect X$

$=$

$\ds \frac \beta {\paren {1 - 0 \beta}^2}$

$\ds $

$=$

$\ds \beta$

$\blacksquare$