Fit distribution to estimate the probability of data less than a value beyond the range of the sample I have a set of data, called $S$, on an interval of values $[a,b]$ (which is not $[0,1]$). I am searching for a distribution that fits the data. I have tried the normal, lognormal, gamma, etc. None of them fit the data well.  
I get the best fit with the beta distribution, after normalizing the data to the interval [0,1] (i.e., $ ( x - a ) / ( b - a)$). Is this a good idea? 
I am interested in the probability of a value: $ c < a$. When I use the normal, I get a very small probability, but different from zero. When I use the beta, I get a probability of 0, because $ c $ is not in $[a,b]$.
Is a good idea use the transformation of data for get a better fit with beta? 
 A: The transformed beta distribution you use is explicitly bounded on the interval $[a,b]$. Any values outside of that interval must have probability 0  by the law of total probability. You are explicitly making the statement that an observation outside of that interval is impossible. Therefore, it's not surprising that when $c \not \in [a,b]$ that the $Pr(c<a)=0$. This was built into your analysis based on the assumptions that you made in modeling the process.
Whatever criterion you used to select the beta model as having the "best fit" did not account for this feature of your data having bounds other than $[a,b]$. Some other model may be more appropriate for the process in question because it will allow for data outside of the interval $[a,b]$, even if the "fit" is worse. Indeed, I would say that modeling data with inappropriate bounds implies a very bad fit in this case.
A: If your distribution is defined to exist on the interval $[a, b]$ where $a$ and $b$ are both known then it is both legal and sensible to consider a beta distribution after scaling to $[0,1]$. 
But the bounds really must apply; you can't at the same time have even one value outside that interval. 
Also, the bounds must be bounds of principle. If either $a$ or $b$ is just a known extreme so that values larger or smaller could occur, then a beta distribution is not appropriate. 
