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Re: random numbers

From: john
Subject: Re: random numbers
Date: Tue, 16 Feb 1999 21:24:43 +0000 (GMT)

Selectively quoted and editted, On Mon, 15 Feb 1999, G. Brger wrote:

> ... is it reasonable that for 2 independant random series, 
> generated with
>    e=normal_rnd(my_mean, my_var, 100000, 2);
> then cor(e(:,1),e(:,2)) has the order of 1e-3?
Yes. Because doing

  e=normal_rnd(my_mean, my_var, n, 2);
  u = cor(e(:,1), e(:,2))

could be viewed as an unusual way to calculate a random number u with 
mean nearly zero, variance roughly 1/n. So we expect u to lie within 
about 3 standard deveations (3/sqrt(n)) of zero. For the n=100000 in the
question, we expect numbers in the range 10e-3 to -10e-3. 
The underlying reason is that evaluating cor(e1,e2) involves averaging 
the product of pairs of independant random numbers, mean 0 variance 1.
Each of the products gives a random number with mean 0 variance 1.
The average then has mean 0, variance 1/n. (Don't quote me - there 
is some detail missing).   
Further numerical evidence could come from a bit of octave code like:

   mu = 200 ;      ## choose some numbers
   sigma = 142 ;

   n = 100 ;       ##  length of each random sequence 
   m = 500 ;       ##  number of correlation coefficients to calculate

   t = zeros(1,m) ;
   for r=1:m
      e = normal_rnd( mu, sigma, n, 2 ) ;
      e1 = e(:,1) ;
      e2 = e(:,2) ;

      t(r) = cor( e1, e2 ) ;

   printf("mean of correlation coefficients is %f\n", mean(t)) ;
   printf("var  of correlation coefficients is %f\n", var(t)) ;
   printf("reciprical of n is %f\n", 1.0/n ) ;

which for me gives values
   mean of correlation coefficients is 0.000501    <- nearly zero
   var  of correlation coefficients is 0.010560    <- nearly 1/n
   reciprical of n is 0.010000                     


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