# Probability distribution of a multiple variable sum

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Rémy Bretin 2019 年 5 月 10 日

Hi everyone,
I’m coming here for really advance statistic/probability advice, which I'm a beginner in this field.
I would like to know the probability of a variable TAU_total such as TAU_total=TAU1+TAU2+….+TAU129.
The variables TAUi are independent of each other.
For each one of them, I have a sample of 20,000 values which you can see some examples of their distribution on the histograms in the attachment.
My question is the following: I would like to be able to determine the probability of TAU_total to be superior to a certain value Xmax.
Regards,
Rémy ##### 4 件のコメント表示非表示 3 件の古いコメント
John D'Errico 2019 年 5 月 11 日
Admittedly, when I first saw this question, I read it as the sum of 12 terms, not 129. 129 terms will cause pretty much anything to look as if it is normally distributed. :)
N = 129;
alph = rand(1,N)*2 + .5;
bet = rand(1,N)*2 + .5;
betamean = alph./(alph + bet);
betavar = alph.*bet./((alph + bet).^2.*(alph+ bet+1));
CLTmean = sum(betamean);
CLTvar = sum(betavar);
CLTstd = sqrt(CLTvar);
nsim = 100000;
X = zeros(nsim,N);
for i = 1:N
X(:,i) = betarnd(alph(i),bet(i),[1,nsim]);
end
MCsum = sum(X,2);
MCmean = mean(MCsum);
MCvar = var(MCsum);
MCstd = std(MCsum);
[CLTmean, MCmean;CLTvar,MCvar;CLTstd,MCstd]
ans =
61.6267682855143 61.6366984653151
7.56366115010423 7.53757246937701
2.75021111009759 2.74546398071018
Comparing the histograms, we see:
histogram(MCsum,100,'norm','pdf')
hold on
fplot(@(x) normpdf(x,CLTmean,CLTstd),[min(MCsum),max(MCsum)],'r') So I don't see any problem using either approach. With only 12 terms in the sum, I'd probably go with the Monte Carlo.

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### 採用された回答

Torsten 2019 年 5 月 10 日

Take samples from your empirical data and use Monte-Carlo-simulation to determine the above probability.
This code should help to take the samples:
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John D'Errico 2019 年 5 月 10 日
Yes.

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### その他の回答 (1 件)

Rémy Bretin 2019 年 5 月 14 日
Thank you everyone for your support.
I decided to go for the MonteCarlo method which gave me a gaussien at the endas you expected.

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