I'm using a uniform random variable (0 mean, 1 variance) to generate white noise. Then I'm doing a moving average on those samples. I'm trying to figure out the variance of this output distribution.

Since I have a fixed number of independently distributed samples, $n$, that are being added together, this feels like the distribution you get from adding die rolls. Unlike a multinomial distrubution, the trial outcomes get added together and are continuous but bounded. (Well, given that it's floating point data, they're actually discrete but large.)

Does this distribution have a name? From a histogram, it looks like a binomial distribution. Using the variance for a multinomial distribution doesn't really work because I'm adding the samples, not treating them as un-addable categories.


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    $\begingroup$ Gaussian distribution. $\endgroup$ – Adrian Keister May 7 '20 at 20:56
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    $\begingroup$ Although it's not Gaussian, a Gaussian becomes an attractive approximation once $n$ exceeds 2. See stats.stackexchange.com/questions/41467 for illustrations and some analysis. But you don't need to know any of this to find the variance, because that depends only on basic properties of variances rather than on the distribution. $\endgroup$ – whuber May 7 '20 at 21:18
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    $\begingroup$ That is good intuition on your part to see the analogy with the binomial distribution. It is interesting to see the role that binomial coefficients play in the Irwin-Hall PDF. $\endgroup$ – John Coleman May 8 '20 at 10:44

The sum of i.i.d. uniform random variables follows the Irwin–Hall distribution.


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