Problem 331. Compute Area from Fixed Sum Cumulative Probability

Created by Roger Stafford

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In Matlab the code<pre> v = rand(1,3);
 v = v/sum(v);</pre>is sometimes suggested as a convenient means of generating three random variables, whose ranges are restricted to [0,1], which have a fixed sum of one. However, this procedure has the property that the area-wise density distribution of the three values, considered as cartesian coordinates in 3D space, is widely variable throughout the planar region of possible locations of v. For any given density value in the range of this density distribution, let A be the corresponding area of the subregion of all points whose density is less than or equal to this given value, and let P be the corresponding probability that v would lie in this subregion. The task is to write a function 'fixedsumarea' which receives P as an input and gives A as an output:<pre> A = fixedsumarea(P);</pre>You should assume that initially 'rand(1,3)' perfectly generates three independent random variables each uniformly distributed on [0,1], but subsequently each is modified by being divided by their mutual sum.

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Last Solution submitted on Sep 10, 2020

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Raphael Cautain on 26 Feb 2012

A really nice problem : concrete origin, careful formulation, then a good walk in 3D geometry, analysis, integration and at the end a single formula. The critical value cuts the triangle in four pieces : my son said it was Zelda !

David Young on 26 Feb 2012

Raphael: I agree completely!

Alfonso Nieto-Castanon on 26 Feb 2012

Perhaps you could have a precision requirement a bit more liberal in order to allow numerical approximation algorithms or other approaches as well? just my two cents, the problem looks great

Rafael S.T. Vieira on 30 Jul 2020

Tip: the probability grows from the vertices of simplex (triforce/triangle) toward its center (which has the highest frequency).

Solution Comments

Solution 256265

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Rafael S.T. Vieira on 30 Jul 2020

This is not a solution, and it should be deleted. And If it is possible, then more test cases should be added.

Solution 50615

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1 Comment

Roger Stafford on 24 Feb 2012

I am pleased that you solved this problem, David. Congratulations! I didn't find any particularly easier way of solving it. The crucial step is showing that the probability density is proportional to your 1/y^3 for points within the corresponding "kite-shaped region". I used the Jacobian between two coordinate systems to show that. After dividing that region into two halves everything falls into place, though in my dotage I had to make heavy use of the Symbolic Toolbox to check for errors. (I hope this problem will serve as a warning to people who recommend this method of producing random numbers with a predetermined sum.) R. Stafford

Solution 47851

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1 Comment

Roger Stafford on 23 Feb 2012

It is inherent in the definition of P here that the density, dP/dA, must increase as P increases and therefore dA/dP must decrease. In your proposed solution you have dA/dP increasing as P increases. R. Stafford

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Problem 331. Compute Area from Fixed Sum Cumulative Probability

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Problem 331. Compute Area from Fixed Sum Cumulative Probability

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