Assuming that we know quantiles 0.25, 0.5 and 0.75, is it possible to fit a distribution from these values ? What distribution ? How ?
Thank you
Edit : I forgot to mention that I would like the distribution to be continuous, like a gamma distribution for instance.
Edit 2 : After reading the comments, I'm realizing that my problem was a bit ill posed. So, to be more precise, my problem is that I've 3 quantiles that define the statistics of a distribution that describes the likelihood of a river streamflow forecast. Usually, to describe such variable, a normal law, a log normal law, a Pearson distribution or a GEV distribution may be used but more frequently a simple gamma law is used. In my particular case, I need a distribution that is skewed as my quantiles 0.25 0.5 and 0.75 are clearly not "equally spaced" and the probability of variable must tend to 0 as it gets further to the median. So my question is twofold: 1) what distribution would fit these constrains and would be appropriate to describe such variable 2) is there a simple way to find the parameters of the distribution that fits my 3 quantiles ?
There's infinitely distributions that would fit those given quantiles. For concreteness, suppose $q_{\frac14}=-1$, $q_{\frac12}=0$, and $q_{\frac34}=1$. Then for example a discrete random variable $X$ with probability mass function $$\mathbb P(X=k) = \begin{cases}\frac14,& k=-1,0,1,m\\0,&\text{otherwise} \end{cases}$$ fits those quantiles for any integer $m\geqslant 2$.