I'm looking for a continuous distribution which I can parameterize such that
- The expected value is roughly zero
- The expected maximum given $x$ draws from that distribution is only very weakly increasing in $x$
- $Prob(\max $ of $ x $ draws $ > 0)$ is small for "large" $x$, say around $x\in (30, 100)$
- (nice to have): The expected maximum of $x$ variables drawn from the distribution has a nice closed-form solution
- (nice to have): The distribution has not too many parameters and is not a very peculiar one that is only known to probability fetishists.
Phenomenon I am trying to capture a phenomenon which I can here describe as a rigged lottery with continuous outcomes. Most lottery tickets are of (varying) negative value. However, as people take part in the lottery, the expected value must be non-negative. Second, there are lottery insiders which, instead of the unconditional value, use insider information to attain the maximum of their draws. These are observed to purchase many tickets. Properties 2 and 3 control that these insiders indeed purchase many tickets (and the value that they expect from that).
Now I try to back out the lottery distribution consistent with this behavior.
I was starting off with Frechet (also because it has convenient properties regarding the maximum), but it doesn't allow enough freedom. I manage to calibrate it such that 1. and 2. hold, but then 3. breaks very quickly.
I looked in the related distributions but couldn't find anything. In general, how does one find (search) distributions when having so specific demands? In specific here, is there any?