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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?

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1 Answer 1

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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 $c>0$:

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

enter image description here

... 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).

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  • $\begingroup$ 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. $\endgroup$
    – whuber
    Commented May 23, 2021 at 19:48
  • $\begingroup$ Yes there should be a simpler form, I wish I could figure it out.... $\endgroup$
    – Aenaon
    Commented May 24, 2021 at 19:53

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