Expectation of Exponential Distribution/Proof 1

Proof
The expectation of a continuous random variable $X$ with sample space $\Omega_X$ is given by:


 * $\ds \expect X := \int_{x \mathop \in \Omega_X} x \map {f_X} x \rd x$

where $f_X$ is the probability density function of $X$.

For the exponential distribution:
 * $\Omega_X = \hointr 0 \infty$

From Probability Density Function of Exponential Distribution:
 * $\ds \map {f_X} x = \frac 1 \beta \map \exp {- \frac x \beta}$

So:
 * $\ds \expect X = \int_0^\infty x \frac 1 \beta \map \exp {- \frac x \beta} \rd x$

where $\exp$ is the exponential function.

Substituting $u = \dfrac x \beta$, we have:


 * $\ds \expect X = \beta \int_0^\infty u \map \exp {-u} \rd u$

The integral evaluates to:


 * $\ds \expect X = \bigintlimits {-\beta \paren {u + 1} \map \exp {-u} } 0 \infty$

So:


 * $\ds \expect X = \beta - \beta \lim_{u \mathop \to \infty} \frac {u + 1} {\exp u}$

By Limit at Infinity of Polynomial over Complex Exponential, it follows that this limit is zero, so that:


 * $\expect X = \beta$