# Expectation of the log shifted-gamma

If $$X \sim \Gamma(a, \beta)$$ and $$c$$ some constant then $$Y=X+c$$ follows a shifted Gamma distribution with pdf $$f_Y(y)=\frac{\beta^a}{\Gamma(a)}(y-c)^{a-1}e^{-(y-c)\beta}$$

$$y\in[c, +\infty)$$. What is the expectation of $$Z = \log Y$$ please?

It appears that the solution is not 'easy'.

Let $$X \sim \text{Gamma}(a,b)$$ with pdf $$f(x)$$, where $$b = \frac{1}{\beta}$$, and assume $$c>0$$:

Then, the desired expectation $$\mathbb{E}[\log(X+c)]$$ is:

... using here the Expect function from the mathStatica add-on to Mathematica, and where links are provided to the definitions of the HypergeometricPFQ and PolyGamma functions.

Numerical checking via Monte Carlo suggests the solution appears correct (avoid discontinuity at integer valued a).

• The numerical checking is wise, because results have have explicitly nonzero imaginary parts are not trustworthy! There must be a simplification that makes the result an obviously real number.
– whuber
Commented May 23, 2021 at 19:48
• Yes there should be a simpler form, I wish I could figure it out.... Commented May 24, 2021 at 19:53