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?
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
2
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).
-
$\begingroup$ 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 ♦May 23, 2021 at 19:48
-
$\begingroup$ Yes there should be a simpler form, I wish I could figure it out.... $\endgroup$– AenaonMay 24, 2021 at 19:53