Transcript Most Random Number - California State University, Stanislaus

Most Random Number
Tom Carter
CSSS
2004
Here’s a somewhat strange
question . . .
What is the most random
number between 1 and
1000?
How to go at it?
Let’s try to develop a “probability
distribution function” representing some
general notion of a “randomness”
property for numbers . . . :-)
First guess: everything is normal, right?
Too simple. We do get a unique maximum
at 500, but that can’t be right.
As everybody knows, you never get the
actual data value. There are always
“errors” that push you away from the real
value.
So, we should overlay a repelling “error
dark-force” on top.
Error dark-force
Overlay the dark-force on the normal
distribution:
Now we’re getting somewhere!
But, we don’t have a unique maximum any
more.
There must be a bias in one direction or
the other, mustn’t there?
Aha! We forgot about Benford’s law on
the distribution of significant digits.
One form of Benford’s law says that on
average, the proportion of numbers having
first digit less than d will be about log(d+1)
for d = 1, 2, . . ., 9.
Another way to say this is that the
distribution will be approximately 1/(d+1),
so we’ll overlay a power law
1/(x/100 + 1)
Normal, with dark-force
and digit overlay
Now we’re almost done. We have a single
maximum, and it is not the simplistic 500.
One more piece. Random numbers must
not have “special properties,” right?
So, they probably won’t be divisible by
small primes like 2 or 3 or 5.
Let’s write a couple of small MatLab
functions for all this.
function r = mrn(x)
%"most random number" function :-)
r = (1/((x/100) + 1)) * (1 - normpdf(x, 500, 70) normpdf(500,500,70))
* normpdf(x, 500, 167);
function r = maxrn
% search for "most random number" using mrn(x)
maxval = 0;
maxrn = 0;
for n = 1:1000
if rem(n,2) ~= 0 & rem(n,3) ~= 0 & rem(n,5) ~= 0 & mrn(n) > maxval
maxval = mrn(n);
maxrn = n;
end
end
r = maxrn;
Now we run our function, and there it is,
the most random number between 1 and
1000:
>> maxrn
ans =
347
What about more general versions of
this?
In other words, what about the most
random number between 1 and 10, 1 and
100, 1 and 10000, etc.?
function r = mrng(x,maxn)
%"most random number" function :-)
r = (1/((10*x/maxn) + 1)) * (1 - normpdf(x, maxn/2,
0.07*maxn)/normpdf(maxn/2,maxn/2,0.07*maxn)) * normpdf(x, maxn/2,
maxn/6);
function r = maxrng(maxn)
% search for "most random number" using mrng(x,maxn)
maxval = 0;
maxrn = 0;
for n = 1:maxn
if rem(n,2) ~= 0 & rem(n,3) ~= 0 & rem(n,5) ~= 0 & mrng(n,maxn) > maxval
maxval = mrng(n,maxn);
maxrn = n;
end
end
r = maxrn;
Summary of most random numbers:
1 to 10:
7
1 to 50: 17
1 to 100: 37
1 to 500: 173
1 to 1000: 347
1 to 10,000: 3,479
1 to 100,000: 34,799
1 to 1,000,000: 347,971