I'm modelling data from a behavioural task. Participants do a few hundred trials. On each trial, they see a sequence of letters at a point on the screen and one of these letters appears surrounded by a white circle. Their task is to report the letter within the circle. Any response they make can be mapped onto a point in time on the trial relative to the white circle because there are no repeats in the sequences that they see. A response of the letter in the circle would have a value of 0; one letter before would have a score of -1; the letter after would have a score of 1 and so on. We've been modelling the distribution of these temporal errors with a mixture of a uniform distribution and some other, non-uniform distribution. Up until now the non-uniform distribution has been Gaussian, but certain theoretical considerations have led us to consider that we need a positively skewed component with a domain that is bounded at zero instead of the Gaussian. I considered using the lognormal distribution, but this is a bad choice because it is undefined at zero.
What positively skewed distributions can model values of zero and greater?
I'm using Matlab. Something that has a PDF written in that language would be great (I'm a scientist, not a statistician).