# The expectation value of the exponential distribution

The probability density function of the exponential distribution is
. The expectation value for this distribution is
.

### Proof

The probability density function of the exponential distribution is
. By definition, the expectation value is
.
By partial integration, the integral of the right-hand side is
. The first term of the right-hand side of $(2)$ is zero, because
, where we used L'Hospital's rule.
Therefore the integral of $(2)$ is
. Substituting this into $(1)$, we obtain
.
Example
The figure below is the exponential distribution for $\lambda = 0.5$ (blue), $\lambda = 1.0$ (red), and $\lambda = 2.0$ (green).
As the value of $\lambda$ increases, the distribution value closer to $0$ becomes larger, so the expected value can be expected to be smaller. In fact, the expected value for each $\lambda$ is
. The larger the value of $\lambda$ is, the smaller the expectation value is.