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August 2nd, 2019, 06:52 AM   #1
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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.
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August 2nd, 2019, 12:41 PM   #2
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C1, etc. are defined as known constants. What do you mean by the mean and std. dev. of C1, etc?
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August 2nd, 2019, 07:27 PM   #3
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For example, [C1,C2,C3,C4] = [1,2,3,4], average=2.75, stdev=1.7
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August 2nd, 2019, 07:41 PM   #4
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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
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August 3rd, 2019, 06:47 AM   #5
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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
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August 3rd, 2019, 01:31 PM   #6
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I found the note too difficult to read.
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August 3rd, 2019, 02:08 PM   #7
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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 $1$ ?
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August 4th, 2019, 01:25 PM   #8
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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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