# Central Limit Theorem

Below is give an excerpt from Central Limit Theorem of CFA level 1 books "

“To illustrate the power of the central limit theorem, we conduct a Monte Carlo simulation to study the capital expenditure plans of telecom businesses. In this simulation, we collect 200 random samples of the capital expenditures of 100 companies (200 random draws, each consisting of the capital expenditures of 100 companies with n = 100). In each simulation trial, 100 values for capital expenditure are generated from the uniform (0, 100) distribution. For each random sample, we then compute the sample mean. We conduct 200 simulation trials in total. Because we have specified the distribution generating the samples, we know that the population mean capital expenditure is equal to (\$0 + \$100 million)/2 = \$50 million; the population variance of capital expenditures is equal to (100 − 0)2/ 12 = 833.33; thus, the standard deviation is \$28.87 million and the standard error is 28.87/√100 = 2.887 under the central limit theorem.”

Could someone explain me pls this expression “(100 − 0)2/ 12 = 833.33,” especially “12” as a denominator?

Thanks

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For a uniform distribution over an interval [a,b], the mean is (a+b)/2 and the variance is (b-a)2/12.

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Got it.

Thanks a lot!

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Was this even part of the material…lol. First time I see this!

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This confused the heck out of me as well for sure. New concepts are usually not introduced in the sample questions, but this case seems different.

this vid explains it on youtube — search " Stats: uniform distribution variance: Why the 12?"

this follows from basic probability theory by taking the difference between E(X^2) and E(X)^2
you do need to know introductory calculus in the derivation, which is beyond the scope of the CFA exams.

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