Re: problem with rand ?

[Paul Kienzle]

Dimitri,

The first thing to note is that I'm only using 32 bits of randomness, 
not 53.
I was wondering if that might be a problem.  Apparently it is.

With a slight change to the code (using rand53() rather than randu())
the following expression usually equals 0:

        length(find(diff(sort(rand(1e6,1)))==0))

Calculating the probability of at least one duplicate is the birthday 
paradox:

        1-p = m!/(m^n (m-n)!)

Using Stirling's approximation,

        log(1-p) ~ (m-n+1/2)(log m - log m-n) - n

For m=2^32, n=1e6,  1-p ~ 2.7e-51   => very high probability of 
duplicates.
For m=2^53, n=1e6,  1-p ~ 0.99983  => very low probability of 
duplicates.
For m=2^53, n=6e7,  1-p ~ 0.55  => even chance of a duplicate.
For m=2^32, n=7.7e4, 1-p ~ 0.50 => even chances of a duplicate.

Unfortunately, my computer is not hefty enough to run this experiment
for n=6e7 (it takes 15 min/trial), but running 100 trials on n=7.7e4
yields a not unexpected 54 trials with no duplicates.  FWIW, the n=6e7
trial I did run yielded 0 duplicates.

The 53-bit code is some 35% slower than the 32-bit code.  Unless
there are objections, I will change to 53-bits.  Accuracy over speed.

- Paul


On Jul 27, 2004, at 11:46 PM, Dmitri A. Sergatskov wrote:

> I found that random series returned by rand (from octave-forge) returns
> a higher number of _identical_ values than I would expect:
>
> octave:1> s1=rand(1,128*1024); ss1=sort(s1); dss1=diff(ss1); 
> any(dss1==0)
> ans = 1
> octave:2> s1=rand(1,128*1024); ss1=sort(s1); dss1=diff(ss1); 
> find(dss1==0)
> ans = [](1x0)
> octave:3> s1=rand(1,128*1024); ss1=sort(s1); dss1=diff(ss1); 
> find(dss1==0)
> ans = 60641
> octave:4> s1=rand(1,128*1024); ss1=sort(s1); dss1=diff(ss1); 
> find(dss1==0)
> ans = [](1x0)
> octave:5> s1=rand(1,128*1024); ss1=sort(s1); dss1=diff(ss1); 
> find(dss1==0)
> ans =
>
>   26998  27604
>
> octave:6> s1=rand(1,128*1024); ss1=sort(s1); dss1=diff(ss1); 
> find(dss1==0)
> ans = 16034
> octave:7> s1=rand(1,128*1024); ss1=sort(s1); dss1=diff(ss1); 
> find(dss1==0)
> ans = 23032
> octave:8> s1=rand(1,128*1024); ss1=sort(s1); dss1=diff(ss1); 
> find(dss1==0)
> ans = 121716
>
> etc...
>
> I would think that for random 2^17 numbers probability of two identical
> numbers would be something like 2^17 / 2^54 (assuming ieee 64 bit 
> double
> representation)
>
> Matlab does not show this behavior even for larger series:
>
> >> s1=rand(1,256*1024); ss1=sort(s1); dss1=diff(ss1); any(dss1==0)
>
> ans =
>
>      0
>
> >> s1=rand(1,1024*1024); ss1=sort(s1); dss1=diff(ss1); any(dss1==0)
>
> ans =
>
>      0
>
> >> s1=rand(1,1024*1024); ss1=sort(s1); dss1=diff(ss1); any(dss1==0)
>
> ans =
>
>      0
>
> Am I missing something here?
>
> Sincerely,
>
> Dmitri.
>
>
>
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