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Can anyone show the sum of two $U(-1,1)$? Any statistical methods to solve the sum of three and more $U(-1,1)$? I am doing a uniform random walk simulation. Can simulate using coding not can not compare with the theoretical values?

COOLSerdash
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shaw shen
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  • Also, see [Irwin-Hall distribution](https://en.wikipedia.org/wiki/Irwin%E2%80%93Hall_distribution). – COOLSerdash Aug 22 '21 at 09:20
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    Convolutions of multiple $U(a,b)$ distributions can be obtained from the convolution of multiple standard uniforms by shifting and scaling (which is easily enough obtained by simply matching the endpoints). Note that sums of many independent and identically distributed uniform random variates will be very well approximated by normal distributions with the same mean and variance, so aside from the first steps of the random walk (a few dozen would easily suffice for most purposes), you can save effort by taking advantage of the Gaussian approximation. – Glen_b Aug 22 '21 at 17:04

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To quickly answer this: The PDF of the sum of $n$ iid $U(-1,1)$ distributed random variables is (for a derivation see here): $$ (1/2)f_X\left(\frac{x + n}{2};n\right) $$ where $f_X$ denotes the PDF of the standard Irwin-Hall distribution which is: $$ f_X(x;n)=\frac{1}{2(n-1)!}\sum_{k=0}^{n}(-1)^k {n\choose k}(x - k)^{n-1}\operatorname{sign}(x - k) $$ where $\operatorname{sign}$ is the sign function. So for example for $n=2$ the PDF is:

$$(1/2)f_X\left(\frac{x + 2}{2};2\right)= \begin{cases} \frac{2 + x}{4}& -2\leq x < 0\\ \frac{2 - x}{4} & 0\leq x\leq 2 \\ 0 & \text{else} \end{cases} $$

COOLSerdash
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