A distribution such that these expectation are 'closed-form'

Question

I am seeking a continuous distribution with real positive support for the random variable $X$ such that, for all $t \in \mathbb R_{+}$,

$$\mathbb E \left(\ln\left(1+tX\right)\right)$$ is given in a 'nice enough' closed form.

My first guesses were an exponential, a lognormal, but nothing worked properly. Of course 'nice enough' is not properly defined... This should be a function of $t$ that is, e.g, not an infinite serie, such that it is easy to compute with a computer.

To sum up, the conditions i have are :

• The distribution must be continuous and positive
• The expression of the expectation must be closed form.

Do you happend to know such a distribution ?

Answer 1

For instance, for $X$ with pdf $f(x)=1(x>0)/(1+x)^2$, by integration by parts, the expectation in question is $t\ln t/(t-1)$ for $t\in(0,\infty)\setminus\{1\}$, and it is $1$ for $t=1$.

More generally, explicit results obtain for any $X$ with a cdf $F$ rational on $[0,\infty)$; for instance, one may take $F(x)=\big(\frac{p(x)}{1+p(x)}\big)^k\,1(x\ge0)$, where $p$ is any nonzero polynomial with nonnegative coefficients and $k$ is a natural number. In the above example, $F(x)=x/(1+x)$ for $x\ge0$.