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A particle randomly hops a discrete distance from one position to another. I have measured, for 200 hops, the time between each hop. Here is the histogram:

enter image description here

What distribution is this?

To look at, it makes physical sense. But, on mathematical grounds, I was expecting it to be exponential.

EDIT. Here is some further information on the actual process:

The data come from a series of molecular simulations of a particle randomly diffusing (random walk) within a free energy minimum (so it's trapped). Occasionally (but rarely) there will be a sufficiently large thermal fluctuation for it to escape the minimum and move (hop) to an identical neighbouring site. It is the time between these hops that I'm measuring.

lemon
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    Do you have a physical model of the hopping? It makes more to sense to start with physics and propose a distribution accordingly. In sociological contexts, I've seen numerous studies of mobility models that propose heavy-tailed distributions ([e.g.](http://www.nature.com/srep/2013/130918/srep02678/full/srep02678.html)) but your particles are not necessarily governed by the same laws as humans. – Emre Nov 22 '14 at 21:05
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    It could be any number of things (it might be consistent with Rayleigh, Gamma, Weibull,... or any number of other possibilities; it might be none of them). More details of the situation may help. Better to choose a model via theoretical understanding than by just guessing from a histogram, which [can be very misleading](http://stats.stackexchange.com/questions/51718/assessing-approximate-distribution-of-data-based-on-a-histogram/51753#51753). Even the best display won't typically allow us to guess correctly. – Glen_b Nov 23 '14 at 02:26
  • I have updated my question. – lemon Nov 24 '14 at 12:58

2 Answers2

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Knowing none of the relevant background, it looks like you may be able to reasonably fit a gamma distribution. (The exponential distribution is a special case of the gamma, in case that has any bearing on the mathematical grounds you mentioned.) I am no particle physicist: Even with the benefit of your edit, I cannot theorize why a gamma distribution would describe your phenomena of interest.

But there are plenty of distributions with support on $(0, \infty)$ and ways to assess fit. If you peruse a few of them, you'll find that many can be fit to give shapes similar to your histogram. Which is best suited could be a matter of assumption, theory, relevant literature, or a question to be answered from the data. Trying to eyeball it is little more than a guess.

To this question in your comment:

Suppose it's not clear which distribution it should follow - is it ever acceptable to just fit a distribution that looks right and draw conclusions from it?

I interpret "looks right" to mean a similarity between the histogram and the plot of an assumed density function. I cannot imagine a case in which this would be well-advised. As the answer Glen references in his comment nicely explains, our interpretation of histograms can be varied and sometimes inaccurate, depending on how they're assembled. An arguably better visual tool for visually exploring fit is a qq-plot. (See this example of them in action.)

All to say, visual tools are great for exploratory analysis, but the conclusions are best left to properly chosen tests. (Or posterior model odds, if Bayesian.)

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Sean Easter
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  • Thank you Sean, it certainly looks like a gamma distribution for a shape parameter of 2 although I'm struggling to see why it should be this particular distribution. I have updated my question with a little more information on the process in case that helps. Suppose it's not clear which distribution it should follow - is it ever acceptable to just fit a distribution that *looks* right and draw conclusions from it? – lemon Nov 24 '14 at 12:55
  • @lemon Edited my answer to address this. Hope it helps! – Sean Easter Nov 24 '14 at 14:48
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It turns out to be a first-hitting-time distribution:

http://en.wikipedia.org/wiki/First-hitting-time_model http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution

lemon
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  • Sorry to go back to such an old thread, but I am just curious why the inverse Gaussian distribution was necessarily the 'right' answer to this question, rather than Sean's first guess of a Gamma distribution. It seems that, with the right parameters, the two distributions can be made to look identical to the eye, and in that case, both fit your histogram almost perfectly. If it's true that it is "never" right to do model selection, i.e. choose a fitting curve, on the basis of what theoretical distribution your observed distribution best reminds you of, then how are we meant to make a choice? – z8080 Aug 16 '16 at 10:44
  • I suppose part of the answer has to do with apriori/theretical expectations of a certain distribution that makes more sense than others, but in some cases, one is hard pressed to come up with one such particularly well-suited distribution. Suppose your histogram represented various participants' scores on a test - one would perhaps have apriori expectations of a normal distribution, but when the distribution is in fact very skewed (as yours is), is it really correct to "force" it into a Gaussian "mold"? – z8080 Aug 16 '16 at 10:46