# Choose probability distribution for times to execute a sequence of moves on a Rubik's Cube

I am writing a software to help me train my speedcubing skills (specifically I want to execute 800 different sequences of moves fast) and there is one subproblem I am struggling with:

What kind of probability distribution can I use to model my times to execute a given fixed sequence of moves? It doesn't have to be perfect, but I want at least some probability distribution that seems reasonable. Here are my observations:

• There is kind of a lower bound. Even if it goes well and I make no mistakes, there is just a limit on how fast my fingers can turn.
• Most of the times are kind of close together, but sometimes I screw up and I am much worse than most of my "normal" times.
• Sometimes (rarely) I screw up very badly and really takes 5 times as long as usually.
• Sometimes it goes really well, but then it is only slightly better than most of my "normal" times.

Does any reasonable distribution come into your minds that I could use to model this?

Clarification: The fact that it is a sequence of moves is irrelevant. I consider each sequence as one individual unit and don't care which moves it consists of. (The reason is that there are so many ways to execute a move with your fingers and it really depends which moves come before and after and it's too complicated to model that. So one sequence of moves is just one unit)

There are 800 such sequences and I assume that they should have the same distribution, but with different parameters.

Note: I posted this on MathOverflow recently, but people told me it was the wrong place, so I came here.

• How about a Gamma distribution (wikipedia.com/en/Gamma_distribution)? If you assume the time to complete each move is iid Gamma, the time to complete $n$ moves is $\textrm{Gamma}(nk,\theta)$, where $k$ and $\theta$ are parameters determining the distribution of times to complete a single move (you might estimate these from some data you collect). – Will Aug 25 '17 at 18:16
• Gammas have lower bounds of zero, so they wouldn't be the first choice to model these times. Why not use the distribution of times you have already observed? Is there any need to fit some mathematical formula to it? – whuber Aug 25 '17 at 19:03
• You slightly misunderstood my question, I added some clarification. I don't really care about the individual moves, one sequence can be considered one unit. And I imagine that the times for each of the 800 sequences uses the same type of distribution, but with different parameters. But apart for that, the Gamma distribution looks great! I will take a look at it. – Lykos Aug 25 '17 at 19:04
• why don't you repeatedly time how long it takes you to do a particular sequence of moves ... then you can plot the empirical pdf of time taken, and see what the distribution looks like, and fit a suitable theoretical model – wolfies Aug 25 '17 at 19:13
• My goal is to do the following: My software gets the total time it took me to execute lots of pairs and triples of these sequences (but not the individual times) and I want to estimate the median for each individual sequence in order to make my software give me the slower ones more frequently. I intended to use a maximum likelihood estimator to estimate the parameters of the distribution given this input data and that would give me the medians. – Lykos Aug 25 '17 at 19:24