Hi!

Executive summary:
I'm looking for a function with the following properties:
f(0) = 0.0
f(1) = 1.0
F(t) - F(0) = 0.5
Help!

Background:
I'm working on a jukebox-like application called FairDJ. To select the
next song to play it gives all known songs a score, then sorts the list
of songs by score, then chooses one at random, biased towards the top
of the list.

The amount of bias in the selection function is something that I'm
optimizing. Biasing too heavily towards the start of the list will
make the song selection good but boring. Biasing too little will vary
the song selection a lot, but people will never get to hear the songs
they actually like.

What I do to select a song is that I take a linear random value between
0.0 and 1.0. Then I transform it to a biased value using f(x). The
biased value is still between 0.0 and 1.0.

So the first two conditions for the differential equation are:
f(0) = 0.0
f(1) = 1.0

Then I'd like to be able to say "I want 50% of the selected songs to be
in the top t fraction of the list". I *think* that could be expressed
thusly:
F(t) - F(0) = 0.5

Does that sound reasonable? Is there a function that matches my
requirements?

Since t doesn't change that often, I could potentially use f(x)=x^(Kt)
and re-compute K numerically every time t changes. But if there's an
algebraic solution I'd prefer using that.

Cheers //Johan

maarten van reeuwijk

Hi Johan,

You may be interested in an exponential distribution:

http://en.wikipedia.org/wiki/Exponential_distribution.

From the ***ulative density distribution you'll find that 50% is where

t = ln(2) / lambda

HTH, Maarten

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Maarten van Reeuwijk dept. of Multiscale Physics
Phd student Faculty of Applied Sciences
maarten.vanreeuwijk.net Delft University of Technology