# [R] precision of rnorm

Prof Brian Ripley ripley at stats.ox.ac.uk
Thu Dec 15 19:29:44 CET 2005

```On Thu, 15 Dec 2005, Thomas Lumley wrote:

> On Thu, 15 Dec 2005, Phineas wrote:
>
>> How many distinct values can rnorm return?
>
> 2^32-1.  This is described in help(Random)

Mot for the default method for rnorm, as it uses two runif's.  The answer
is somewhere in the 2^50s, as the base uniform random number uses 2^59 but
some will be mapped to the same result.

>> I assume that rnorm manipulates runif in some way, runif uses the Mersenne
>> Twister, which has a period of 2^19937 - 1.  Given that runif returns a 64
>> bit precision floating point number in [0,1], the actual period of the
>> Mersenne Twister in a finite precision world must be significantly less.
>
> No. Not at all.  Consider a sequence of 1-bit numbers: individual values
> will repeat fairly frequently but the sequence need not be periodic with
> any period
> 1101001000100001000001
> is the start of one fairly obvious non-periodic sequence.
>
> There are reasons that a longer period than 2^32 is useful.  The most
> obvious is that you can construct higher-resolution numbers from several
> runif()s.

And the default method for rnorm does so.

> The Mersenne Twister was designed so that quite long
> subsequences (623 elements) would be uniformly distributed.
>
> Less obvious is that fact that a periodic pseudorandom sequence is likely
> to show a frequency distribution of repeat values that differs from the
> random sequence once you get beyond about the square root of the period.
> This means that a 32-bit PRNG should really have a period of at least
> 2^64.
>
> The randaes package provides a runif() that uses 64 bits to construct a
> double, providing about 53 bits of randomness.
>
>> One of the arguments for Monte Carlo over the bootstrap is that for a sample
>> size n the bootstrap can return at most 2^n distinct resamples, however for
>> even for relatively small sample sizes there may be no increase in precision
>> in using Monte Carlo.
>
> I don't get this at all. What technique are you comparing to the bootstrap
> and for what purpose?
>
> 	-thomas
>
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--
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595

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