# need explanation about the exponent parameter s in zipf distribution

I need to model the popularity of some requested files from a library with Zipf distribution and I want to simulate it in MATLAB. I don't know what's the effect of parameter s on my result. for example, if I choose it greater than 1 or less than one how it can change my result? I've been reading that this parameter determines the shape of the distribution but I don't know what does it really mean. This parameter is something that I should choose in my simulation or it depends on the data that I model?

In every case the probabilities form a decreasing sequence but the rapidity with which they decrease depends on the parameter.

Lower values of the index parameter imply relatively more of the probability is associated with larger numbers ('heavier tail').

If $$s\leq 1$$ then the range must be finite; if $$s>1$$ its possible to have a distribution with no upper limit.

This parameter is something that I should choose in my simulation or it depends on the data that I model?

That depends on what you're trying to do. If you're trying to model data then you would (nearly always) estimate it from data; if you're trying to do something with simulation, it would depend on the purpose of the simulation.

• Thank you so much for your explanation. So if I understand it well, in the case of modeling popularity of requested files from a server, lower values of this parameter cause to have more of unpopular events in compared to popular ones (It's like we say if we increase the parameter, the popular files more likley to happen in our result ) right? If I estimate s near to 1 how do I know this is a good estimation for my case? how do I kmow I should not increase it more? – Bonnie Nov 23 '18 at 15:36
• With random sampling from a population of interest, any estimate will have sampling error. Fewer data points mean greater error. Even with the best possible estimate (for some defined 'best') then of course the 'true' parameter would be higher or lower than a given estimate. You can estimate uncertainty (if the model is correct, how much variation from the 'true' value you'd expect); and you can compute corresponding interval estimates for various quantities (including for predictions). – Glen_b Nov 23 '18 at 23:01
• Ultimately, you need to examine what you're trying to achieve and attempt to do the best that it's possible to do for that situation. – Glen_b Nov 23 '18 at 23:33