For example, if I know the 25th percentile (or the first quartile) of some distribution to be 30 and the 75th percentile (third quartile) to be 70, is it possible to reproduce the underlying distribution?
My first guess was using Maximum Likelihood Estimators to figure out the most likely parameters for the distribution's cumulative distribution function (CDF) since these known values basically mean I know P(X<=30) = 0.25 and P(X<=70) = 0.75. However, I was not sure where to go from there.
Any help or pointers would be much appreciated! Thanks!