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 August 2nd, 2019, 06:52 AM #1 Newbie   Joined: Jul 2019 From: Mumbai Posts: 18 Thanks: 0 Standard deviation on scaling of random variables I have the answer already (based on simulations), but want to know exactly which theorem/ law is at play: Suppose X1, X2, X3, X4 are 4 positive random variables such that X1+X2+X3+X4=1. Suppose I scale them with known constants C1, C2, C3, C4, respectively, then look at the function below F=C1.X1+C2.X2+C3.X3+C4.X4 If we calculate 100000 instances of the above function F (each for a different Xi set), then the mean of F is equal to the mean of C1, C2, C3, C4. I found based on a huge number of simulations that the standard deviation of F is equal to the standard deviation of C1, C2, C3, C4 divided by sqrt(pi*4), where pi=3.142 If the sample size is of 12 numbers F=C1.X1.+...C12.X12, then the standard deviation of F is stdev(C1,...C12)/sqrt(pi*12) So, the formula standard deviation of scaling constants/sqrt(pi.n), any idea if there is such a formula in statistics, because it seems to be true for a lot of simulations. August 2nd, 2019, 12:41 PM #2 Global Moderator   Joined: May 2007 Posts: 6,806 Thanks: 716 C1, etc. are defined as known constants. What do you mean by the mean and std. dev. of C1, etc? August 2nd, 2019, 07:27 PM #3 Newbie   Joined: Jul 2019 From: Mumbai Posts: 18 Thanks: 0 For example, [C1,C2,C3,C4] = [1,2,3,4], average=2.75, stdev=1.7 August 2nd, 2019, 07:41 PM #4 Newbie   Joined: Jul 2019 From: Mumbai Posts: 18 Thanks: 0 In the below link, you can find the excel calculation: Columns A to D are the random variables whose sum is 1. E to H multiply Xi by Ci (for this case, Ci=2,15,32,90. I is the sum of Ci.Xi. K is simply I arranged in ascending order. You can find the standard deviation and average of column K. Stdev of K (10. agrees with my hypothesis: Stdev(Ci)/sqrt(n.pi) = 38.3/3.5=10.8. I generated X1...X4 by RAND() function - generating 4 numbers and dividing each of them by the sum of the 4 numbes. https://drive.google.com/file/d/1BuV...ew?usp=sharing August 3rd, 2019, 06:47 AM #5 Newbie   Joined: Jul 2019 From: Mumbai Posts: 18 Thanks: 0 I tried to formulate with the limited knowledge of variance of scaled sum of random variables that I developed while reading statistics books. The red circled parts are the ones estimated by excel data. They surely hold for n=4 to 100. Please see the below link for the derivation and let me know your comments. I am sure there will be a theoretical basis for the red circled parts: https://drive.google.com/file/d/1-KM...waDDO4D-K/view August 3rd, 2019, 01:31 PM #6 Global Moderator   Joined: May 2007 Posts: 6,806 Thanks: 716 I found the note too difficult to read. August 3rd, 2019, 02:08 PM #7 Senior Member   Joined: Sep 2015 From: USA Posts: 2,529 Thanks: 1389 The whole premise of this problem has issues. You say that random variables $X_{1-4}$ sum to $1$ Sum to $1$ in what sense? If they truly sum to $1$ then at least one of them isn't truly random, but a function of particular instantiations of the others that are. Do you mean the expectation of the sum is $1$ ? August 4th, 2019, 01:25 PM #8 Global Moderator   Joined: May 2007 Posts: 6,806 Thanks: 716 If the Xk all have the same distribution and the sum of the Xk has an average = 1, then getting the mean for F is simple. E(F)=C1+C2+Cc3+C4. To get the variance you need to know the variances for each Xk as well as their dependence on each other if any. Tags deviation, random, scaling, standard, variables 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 John Travolski Probability and Statistics 1 July 14th, 2016 01:21 PM ybot Probability and Statistics 1 August 7th, 2014 10:19 AM messaoud2010 Probability and Statistics 1 July 18th, 2014 11:26 AM mikey Advanced Statistics 1 August 9th, 2012 03:43 AM Axel Algebra 2 April 28th, 2011 03:25 AM

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