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I have multiple IID (bigger then 0) of unknown distribution and I get the max of them as a result of my calculation.

Can I say that the max also have certain gaussian shape distribution? If not what is the distribution of the result?

I saw this answer which describe the result of IID product in a log-normal distribution. but as far as I understand this is not the same as max.

I tried to run a simulation in MATLAB with 1000*1000 uniform IID in range 1000 and got that the max of each column is not distibuted randomlly but getting close to 1000. here is the code:

r = randi(1000,[1000,1000]);
mxR = max(r);
figure;
hist(mxR);

After that I tried some other non - caped distribution (normal dist. with \mu = 5000 and \sigma = 500) and I got something similar tolog-normal dist. distribution result of max IID (normal)

here is the code:

r = random('Normal',5000,500,[10000,10000]);
mxR = max(r);
figure;
hist(mxR);

Is there a rule or a theorem regarding this?

thanks for your help!! (If you think this is interesting (or not) please let me know)

user158881
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  • Hello! Perhaps this is of help: https://stats.stackexchange.com/questions/18433/how-do-you-calculate-the-probability-density-function-of-the-maximum-of-a-sample Have you researched any other material? – Fede Poncio Dec 30 '20 at 14:13
  • thanks for the replay! I'll look into it. I tried and found on google some min-max normal distribution papers which is not relevant to me. also my real distribution is not "normal dist." but something else – user158881 Dec 30 '20 at 14:19
  • @FedePoncio the reference was indeed very helpful. thanks! – user158881 Dec 31 '20 at 07:36

1 Answers1

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Yes, it’s called the Generalized Extreme Value Distribution .

The conditions of convergence are given by the Fisher-Tippet-Gnedenko Theorem.

Basically, it says the distribution of an extreme order statistic (max,min) of an iid sample can converge to one of three distributions.