# Expected log value of noncentral exponential distribution

Suppose $X$ is non-central exponentially distributed with location $k$ and rate $\lambda$. Then, what is $E(\log(X))$.

I know that for $k=0$, the answer is $-\log(\lambda) - \gamma$ where $\gamma$ is the Euler-Mascheroni constant. What about when $k > 0$?

• Have you tried integrating in Mathematica?
– user10525
Jun 18, 2012 at 16:01
• I assume $k > 0$ (when the density is written as $\lambda \exp\{-\lambda(x-k)\}$,) otherwise $x < 0$ with probability > 0, with dreadful consequences for $\mathbb{E}\log x$. Jun 18, 2012 at 16:25
• I got ${\mathbb E}[\log(X)]=e^{k\lambda}\Gamma(0,k\lambda)+\log(k)$. Mathematica is faster if you use the command Assumptions for specifying the parameter space.
– user10525
Jun 18, 2012 at 16:35
• Does the upper incomplete gamma function count as closed form? (To me, it does not.) This is just conveniently hiding an integral via notation. Jun 18, 2012 at 16:50
• @NeilG This is the Mathematica code Integrate[Log[x + k]*\[Lambda]*Exp[-\[Lambda]*x], {x, 0, \[Infinity]}, Assumptions -> k > 0 && \[Lambda] > 0]. You can just copy it and paste it on a .nb file. I am not sure if the Wolfram Alpha allows for including restrictions.
– user10525
Jun 18, 2012 at 18:45

The desired integral can be wrestled into submission by brute-force manipulations; here, we instead try to give an alternative derivation with a slightly more probabilistic flavor.

Let $X \sim \mathrm{Exp}(k,\lambda)$ be a noncentral exponential random variable with location parameter $k > 0$ and rate parameter $\lambda$. Then $X = Z + k$ where $Z \sim \mathrm{Exp}(\lambda)$.

Note that $\log(X/k) \geq 0$ and so, using a standard fact for computing the expectation of nonnegative random variables, $$\newcommand{\e}{\mathbb E}\newcommand{\rd}{\mathrm d}\renewcommand{\Pr}{\mathbb P} \e \log(X/k) = \int_0^\infty \Pr(\log(X/k) > z)\,\rd z = \int_0^\infty \Pr(Z > k(e^z - 1)) \,\rd z \>.$$ But, $\Pr(Z > k(e^z -1)) = \exp(-\lambda k(e^z - 1))$ on $z \geq 0$ since $Z \sim \mathrm{Exp}(\lambda)$ and so $$\e \log(X/k) = e^{\lambda k} \int_0^\infty \exp(-\lambda k e^z) \, \rd z = e^{\lambda k} \int_{\lambda k}^\infty t^{-1} e^{-t} \,\rd t \>,$$ where the last equality follows from the substitution $t = \lambda k e^z$, noting that $\rd z = \rd t / t$.

The integral on the right-hand size of the last display is just $\Gamma(0,\lambda k)$ by definition and so $$\e \log X = e^{\lambda k} \Gamma(0,\lambda k) + \log k \>,$$ as confirmed by @Procrastinator's Mathematica computation in the comments to the question.

NB: The equivalent notation $\mathrm E_1(x)$ is also often used in place of $\Gamma(0,x)$.

• +1 @Michael Chernick It seems like not everyone is lazy ;).
– user10525
Jun 18, 2012 at 18:39
• This is really great. I just want to point out for anyone implementing this that many implementations of the incomplete gamma function restrict the first parameter to be strictly positive. The identity $\Gamma(0, z) = -\operatorname{Ei}(-z)$ solves that minor problem. Jun 18, 2012 at 18:51