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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/1KM...waDDO4DK/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 $X_{14}$ 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.


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deviation, random, scaling, standard, variables 
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