 My Math Forum Central Limit Theorem for weighted summation of random variables?

 July 28th, 2019, 10:20 PM #1 Newbie   Joined: Jul 2019 From: Mumbai Posts: 18 Thanks: 0 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, July 29th, 2019, 03:16 AM #2 Newbie   Joined: Jul 2019 From: Mumbai Posts: 18 Thanks: 0 I forgot to add that all Xi are positive, and sum of all Xi is always 1. 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 CLTA01.jpg (13.0 KB, 3 views) CLTD01.jpg (11.7 KB, 2 views) July 29th, 2019, 09:11 AM #4 Newbie   Joined: Jul 2019 From: Mumbai Posts: 18 Thanks: 0 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 Tags central, limit, random, summation, theorem, variables, weighted Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post bml1105 Probability and Statistics 7 August 21st, 2014 11:24 AM NeedToLearn Advanced Statistics 1 June 21st, 2013 01:26 PM safyras Algebra 0 May 25th, 2011 12:52 PM billiboy Advanced Statistics 3 July 10th, 2009 10:34 AM xoxo Advanced Statistics 0 May 22nd, 2009 10:38 PM

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