Please excuse my naïvety in this matter, but what kind of distribution is this, where the max/min is bounded by a definite interval (in this case, $[0,3]$ )? I can see the (elegant) accepted answer is a triple intergal, but is it possible to define the relevant distribution plot, and integrate once?
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If $X, Y, Z$ are i.i.d. with uniform distribution on the interval $[0,1]$, then $T = X+Y+Z$ has pdf
$$ f(t) = \cases{ {t}^{2}/2&$0 \le t\le 1$\cr -3/2-{t}^{2}+3\,t&$1 \le t\le 2$\cr 9/2-3\,t+{t}^{2}/2&$2 \le t\leq 3$\cr 0& otherwise\cr}$$
martin
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Robert Israel
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must it be defined piecewise? – martin Sep 24 '15 at 20:49
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1Essentially yes (though you could use Heaviside functions etc to get the same effect). The second derivative of the pdf is $+1$ on the intervals $(0,1)$ and $(2,3)$, but $-2$ on $(1,2)$. – Robert Israel Sep 24 '15 at 20:57
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how did you find this? How would I go about finding the pdf for any $X_1, X_2, \dots X_n$ that are i.i.d. with uniform distribution on the interval $[0,1]?$ Where should I look for literature on the subject? – martin Nov 03 '15 at 20:07
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1Actually, I used Maple. But you can do it yourself. If $f_n$ is the pdf for $X_1 + \ldots + X_n$, then $$f_{n+1}(x) = \int_{0}^1 f_n(x-t); dt$$ Do it on each interval $[j,j+1)$ separately. – Robert Israel Nov 03 '15 at 21:30
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thanks - will give it a go - any idea how to set up in Mathematica? – martin Nov 03 '15 at 21:33
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1Sorry, I don't use Mathematica. – Robert Israel Nov 03 '15 at 21:38
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don't worry, thanks for the tip anyway - will give it a go manually :) – martin Nov 03 '15 at 21:40
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For reference: the PDF presented here can be seen as a suitably scaled quadratic B-spline basis function. – J. M. ain't a mathematician Nov 04 '15 at 08:24