# explicit random cake cutting

I like to split a given interval, let's say $[0,1]$, randomly to a given number $n$ parts.

A random input may be provided, like for example a sequence of random numbers $\omega=(r1, r2, ...)\in\Omega$, as usual random number generators generate.

Is there any efficiently computable function $F_\omega(i), i\in 0,..,n$ that gives a random partition with $$F_\omega(k)=\sum_{i\in 0..k} f_\omega(i)$$ so that $F_\omega(n)=1$ and $F(0), ..., F(n)$ will distribute in a evenly fashion, eg. $F_\omega(i)$ with randomly choosen $i$ will have a uniform distribution?

Like a cake cut at $n$ randomly choosen angles...

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I'm not sure what you're after. In terms of the interval, are you trying to generate $n$ random variables that sum to 1 and all have identical uniform marginal distributions? – soakley Nov 19 '13 at 3:13
Yes. However, the variables $F_\omega(k)=\sum_{i\in 0..k}f_\omega(i)$ with $k<=n$ would be even more useful, as they are sorted. But even harder to compute I think... Computing several $f_\omega(i)$ repeatedly to get some $F_\omega(k)$ is not quite effective... Would be cool to have $F$ in an explicit fashion without the need to compute the $f$'s at all. – dronus Nov 19 '13 at 8:49
Edtited question to ask for $F$ instead of $f$. – dronus Nov 19 '13 at 9:05
A use for this would be a fully functional algorithm to create realitstic divisions of exisiting ressources like population to cities or land to countries. Without this functional paradigm, the random distribution would need iterative generation and would'nt fit parallel computation well this way. – dronus Dec 17 '13 at 17:23
For large numbers, n random variables with uniform distribution divided by n would approx. sum of 1 of course... but for smaller numbers this is not feasable. – dronus Dec 17 '13 at 17:25