Differential Entropy of Exponential Distribution

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

Then the differential entropy of $X$, $\map h X$, is given by:


 * $\map h X = 1 + \map \ln \beta$

Proof
From the definition of the exponential distribution, $X$ has probability density function:


 * $\displaystyle \map {f_X} x = \frac 1 \beta e^{-\frac x \beta}$

From the definition of differential entropy:


 * $\displaystyle \map h X = -\int_0^\infty \map {f_X} x \map \ln {\map {f_X} x} \rd x$

So: