# Distribution from quantile data or custom distribution

I'd like to fit a distribution (any you like) based on these requirements:

1. Produces integer values (preferable but not required)
2. Mean ($\mu$)=100
3. Std=114
4. Quantiles( 25%, 50%, 75%)=(6,39,200)
5. Min=0; Max=~300 (but $\infty$ is acceptable);

Poisson fits criteria 1,2 and nearly 3, but not 4 by far. Lognormal... maybe

EXTRA info: value 0 is produced 8% of the time and 300 is produced 18%.

Is it possible to do something like this?

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What you do is maximize $-\sum_i p_i \ln p_i$ under the constraints you have given and where the $p_i$ are the probabilities of all discrete events $i=0,\ldots,300$. You'd best do this numerically.
Do I know all the $p_i$ values though a-priori? We worked this out in a thread here: stats.stackexchange.com/questions/21638/distribution-fitting/… – HCAI Feb 10 '12 at 12:45