Expectation of Exponential Distribution

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Theorem

Let $X$ be a continuous random variable of the exponential distribution with parameter $\beta$.


Then the expectation of $X$ is given by:

$E \left({X}\right) = \beta$


Proof

The expectation is:

$\displaystyle E \left({X}\right) := \int_{x \mathop \in \Omega_X} x \ f_X \left({x}\right) \ \mathrm d x$

which, for the exponential distribution: is:

$\displaystyle E \left({X}\right) = \int_0^\infty x \frac 1 \beta \exp{\left(- \frac x \beta \right)} \ \mathrm d x$

where $\exp$ is the exponential function.

Substituting $u = \dfrac x \beta$:

$\displaystyle E \left({X}\right) = \beta\int_0^ \infty u \exp \left({-u}\right) \ \mathrm d u$

The integral evaluates to:

$\displaystyle E \left({X}\right) = \left.{ -\beta \left({u+1}\right) \exp \left({-u}\right) }\right|_0^\infty$

This evaluation is the limit:

$\displaystyle E \left({X}\right) = \beta - \beta \lim_{u \to \infty} \frac {u+1} {\exp u}$


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

$E \left({X}\right) = \beta$

$\blacksquare$