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So I have been reading about the Irwin-Hall distribution online, it is a sum of uniform distributions on $[0,1]$, and it seems very interesting:

http://en.wikipedia.org/wiki/Irwin%E2%80%93Hall_distribution

On the Wikipedia article above they derive pdf for the special cases n = 1,2,3,4, and 5. n = 1 is trivial, for n = 2 we are drawing points from a square $U_1 \times U_2$. Then we compute the probability of picking points under a line in that square, take its derivative and then derive the triangular distribution. For n = 3, we are drawing points from the cube $U_1 \times U_2 \times U_3$ and compute the probability of picking points under a plane and can intuitively see the parabolic distribution (since the volume under the plane on the cube will be of third degree, then we takes its derivative to get pdf).

In general we see the pdf will have n pieces with each piece of degree k-1. My question here is deriving the formulas for the n = 3,4,5 case don't seem easy as they are shown in the Wikipedia article, what would be the approach to get the equations? Secondly, intuitively we would think due to the central limit theorem that this distribution approaches the normal distribution, but they give a general version of the pdf in the Wikipedia article, and I don't see how that is going to converge to the normal distribution.

For n = 3, I know how to get pdf when $x \in [0,1]$ and $x \in [2,3]$, but not sure about when $x \in [1,2]$.

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The best and simpler way to derive the distribution of the sum of two independent Uniform random variates is geometrical requiring working out some area calculations in 2-D.

However, the derivation of the distribution of $\sum_{i=1}^{n}X_i$ for $n>2$ with each $X_i$ independently distributed as $U(0,1)$ is generally tedious. Geometrically it is difficult to visualise for higher values of $n$. However, a convolution approach would be used to find them (that I will use here kind of recursively).

I start assuming you know the distribution of $A=X_1+X_2$ given by the below pdf: $$f_A(a) = \begin{cases} a & \text{if $0 \le a \le 1$}\\ 2-a & \text{if $1 \le a \le 2$}\\ 0 & \text{elsewhere}\end{cases}$$


For $n=3$, define $B=X_1+X_2+X_3=A+X_3$. Note $0\le B\le3$. Now, by convolution of pdfs, the pdf of B: $f_B(b)=\int_{-\infty}^\infty f_{X_3}(x_3)f_A(b-x_3)\,dx_3$

Note-1: $f_A(b-x_3)=b-x_3$ for $0\le b-x_3\le1$, i.e., $b-1\le x_3\le b$; Also, $0 \le x_3 \le 1$. Combining these two gives $\mathbb{max}(b-1,0) \le x_3\le \mathbb{min}(b,1)$

Note-2: $f_A(b-x_3)=2-b+x_3$ for $1\le b-x_3\le2$, i.e., $b-2\le x_3\le b-1$; Also, $0 \le x_3 \le 1$. Combining these two gives $\mathbb{max}(b-2,0) \le x_3\le \mathbb{min}(b-1,1)$

Looking at the bounds of $x_3$, it is reasonable to break the range of $b$ (i.e.,[$0,3$]) into $0\le b\le1$, $1\le b\le2$ and $2\le b\le3$ and considering the form of the pdf of B within each range separately.

Case: $0\le b\le1$: Range in Note-1 reduces to $0\le x_3\le b$; while that in Note-2 doesn't reduce to a feasible bound for $x_3$. Thus the pdf of $B$ reduces to

$f_B(b)=\int_0^b (b-x_3)\,dx_3=\frac{b^2}{2}$

Case: $1\le b\le2$: Range in Note-1 reduces to $b-1\le x_3\le 1$; while that in Note-2 reduces to $0\le x_3\le b-1$. Thus the pdf of $B$ reduces to

$f_B(b)=\int_{b-1}^1 (b-x_3)\,dx_3+\int_0^{b-1}(2-b+x_3)\,dx_3=\frac{-2b^2+6b-3}{2}$

Case: $2\le b\le3$: Range in Note-1 doesn't reduce to a feasible bound for $x_3$; while that in Note-2 reduces to $b-2\le x_3\le 1$. Thus the pdf of $B$ reduces to

$f_B(b)=\int_{b-2}^1(2-b+x_3)\,dx_3=\frac{(3-b)^2}{2}$


Can you now try similar logic for $n=4$ and $n=5$?

Although it may not be readily intuitive that the general form of the pdf of the sum of Uniform variates would tend to follow approximately Normal for large n, the pdf is a piecewise polynomial function of degree n-1. And, if you plot this function you'll end up with a plot of close to a pdf of Normal distribution (I tried in R) and it indeed goes to show how powerful the CLT is!

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