Re: [Phys-l] quadratic uncertainty

[ludwik kowalski <kowalskil@mail.montclair.edu>]

Dear Brian,

I do not understand the format of your display.

Prompted by you, I wrote a standard Monte Carlo code. I also assumed  
that the uncertainties represent two standard deviations.

I assumed that x must be a real number. But I counted how often the  
calculated x was a complex number.

Here are my results, for 10000 cases


1) when I use + to calculate x

   complex solutions  occurred in 41.59% of cases
   mean x was 1.1156, stand. deviation was 0.9420


2) when I use - to calculate x
   complex solutions  occurred in 40.92% of cases
   mean x was 0.9670, stand. deviation was 0.8053

Suppose I assume that x represent the mass. In this case real but  
negative x must also be ignored. I did not count them but this can  
easily be done. (The percentage additional rejections of would be  
large, as one can guess by looking at standard deviations.

It is too late; good night.

Ludwik

 ===========================================

On Aug 28, 2010, at 6:05 PM, brian whatcott wrote:

>  Here's what Matlab made of the puzzle. It was first necessary to
> decide what meaning to apply to the uncertainty numbers associated  
> with
> the parameters, and what form of that uncertainty to select. I did not
> presume to link the variability between the three parameters.
>  I arbitrarily chose the uncertainty to represent a rather high
> confidence of capturing the variability so I specified the uncertainty
> as two SDs of a normal distribution, and took a hundred samples of  
> each
> parameter. Matlab has a root extractor built in, and it provided  
> results
> as complex numbers, which plot  as red 'x' symbols in a circle in the
> complex plane.
>
>  These are the four lines I entered on the command line.
> > > a=normrnd(1,0.00005,[1 100]);
> > > b=normrnd(-2.08,0.005,[1 100]);
> > > c=normrnd(1.08, 0.005,[1 100]);
> > > plot(roots([a b c]),'xr');
>
> This is the plot depiction as a jpg on a free image server.Forgive  
> the ads.
>
> <http://i880.photobucket.com/albums/ac6/betwys/quadratic/ 
> quadroots.jpg>
>
> Surprising...
> Brian W
>
> $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
> Here's a little puzzle with some seasonal relevance:
>
> We need to find a good value for x
> /and for the uncertainty associated with x/
> given that:
>      a x2  + b x + c = 0         [1]
>      a =  1 ± .0001
>      b = -2.08 ± .01
>      c =  1.08 ± .01
>
> This was mentioned in connection with the annual "sig figs"
> donnybrook on the chemistry list.  There are a thousand people
> on that list, and so far nobody has come up with a solution.
> One person came kinda close, but no cigar.
> $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
>
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Ludwik

http://csam.montclair.edu/~kowalski/life/intro.html