Expectation of Exponential Distribution/Proof 1

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:

$\expect X = \beta$

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:

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

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

$\expect X = \beta$

$\blacksquare$