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July 28th, 2019, 10:20 PM   #1
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Central Limit Theorem for weighted summation of random variables?

Here is a quick question:-

If X1, X2, X3,.... X20 are 20 random variables (independent/ idd)

What would be the result of:

2*X1+5*X2+1*X3+18*X4...+0.5*X20? (linear combination of the random variables, with fixed known constants). Will the above function form a normal distribution if we take 100000 instances of the values of [X1,X2,.....X20]. And will the normal distribution be tighter (smaller variance) if we take 30 instead of 20 random variables?

Does it have to do with Central Limit Theorem? Because I thought CLT was only for arithmetic mean of random variables and not for linear combination (weighted summation/ scaled summation)?

Please help me.

Thanks,
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July 29th, 2019, 03:16 AM   #2
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I forgot to add that all Xi are positive, and sum of all Xi is always 1.
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July 29th, 2019, 08:55 AM   #3
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I ran some simulations for four cases of coefficients (I called them Ci).
Case A: C = {2 5 1 18 3.5 7 7 5 15 2 1 2.5 12 16 13 1 3.5 1.5 0.5 0.5}
Case B: C = {1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20}
Case C: C = {0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 9.6 9.7 9.8 9.9 10.0 10.1 10.2 10.3 10.4 10.5}
Case D: C = {1 1 1 1 1 1.5 1.5 1.5 1.5 1.5 2 2 2 2 2 10 10 10 10 10}

For each case, I ran ten simulations of 100000 tests each and plotted the results. I generated the X values the following way:
Let xi be 20 random values pulled from a flat distribution on [0, 1).
Let Xi = xi/(x1 + x2 + … + x20)

In each simulation, the distribution of the 100000 sums looked superficially like a normal distribution (although at N=20, a normal and a binomial distribution are hard to tell apart).

Also in each simulation, as expected, the mean of the sums was very close to the mean of the Ci coefficients.

A little bit more of a mystery for the moment, in each simulation, the standard deviation of the 100000 sums was very close to 13.2 % or 13.3 % of the population standard deviation of the 20 Ci coefficients.

I may think on that last point for a while, since I do teach introductory Gaussian distribution statistics in one of my classes and it may be insightful to figure out how to predict that.
Attached Images
File Type: jpg CLTA01.jpg (13.0 KB, 3 views)
File Type: jpg CLTD01.jpg (11.7 KB, 2 views)
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July 29th, 2019, 09:11 AM   #4
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Thanks a lot, it was very insightful. On the question of whether there is a theoretical basis, the closest I could reach on Wikipedia was Martingale's version of Central Limit Theorem, but owing to my lack of background in statistics, I could not figure out if it is a real theoretical justification
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