Dear D. W. Ryu,
I'm afraid this is rather beyond my competence. I am therefore
copying your response to the edstat list, in hopes that someone there can
be of more help to you.
On Sat, 15 Jan 2000, D. W. Ryu wrote:
> Dear Dr. or Prof. Burrill,
>
> Thanks for your comments.
> The domain of random variable X and Y is -1< X, Y <1, which is points
> in xy plane. The points is located clustring near origin (0,0), so I
> try to approximate the its density to bivariate normal distribution.
Ah. That explains why (1 - sigma_max*sigma_min) would not be imaginary.
It is still unclear to me why such an expression might approximate the
bivariate correlation rho, however. (That rather contradicts my sense of
the fitness of things, but I cannot supply a good reason for this.)
It seems to me, intuitively, that the boundedness of (X,Y) implies that
the bivariate normal density function may sometimes not be a good
approximation at all, unless the variances are really quite small _and_
the correlation is not close to 1 (or -1).
> To define normal distribution, I need to know three parameters.
> I could define the elliptical probability contour function by
> parameter sigma_Max, sigma_Min, and rotation angle Omega from reference
> axis to semi-axis. The domain of sigma_Max and sigma_Min is the
> distance from origin in generic coordinate.
It is not clear to me what sigma_Max and sigma_Min are; I'd assumed they
were standard deviations of X and Y, which is consistent with your need
for three parameters (bivariate normal would normally require 5
parameters: mean and variance (or s.d.) of each variables, and
covariance (or correlation) between them; but apparently your means are
both zero?
> Can I get the three parameters sigma_X, sigma_Y and correlation
> coefficient from these information.
I'm not sure what information you refer to. You seem already to have
sigma_Max and sigma_Min; I would expect the correlation to be derivable
from the reference angle Omega, but don't know how to do it.
> I am sorry that I didn't inform about question.
>
> Would you please let me know the relationship.....
>
> Thanks in advance.
>
> With my best regards,
>
> D.W. Ryu
>
>
> ----- Original Message -----
> From: Donald F. Burrill <[EMAIL PROTECTED]>
> To: D.W. Ryu <[EMAIL PROTECTED]>
> Cc: <[EMAIL PROTECTED]>
> Sent: Saturday, January 15, 2000 9:44 PM
> Subject: Re: Q: correlation coefficient in bivariate normal distribution
>
>
> > On Fri, 14 Jan 2000, D.W. Ryu wrote:
> >
> > > In bivariate normal distribution of random variable X and Y, is it true
> > > that the correlation coefficient, rho, is equivalent to
> > > sqrt(1 - sigma_Max * sigma_Min),
> > > where sigma_Max is the maximum standard deviation
> > > and sigma_Min is the minimum standard deviation?
> >
> > Maximum and minimum in what domain of discourse? You've mentioned only
> > two variables, which would seem to imply only two sigmas.
> >
> > The relationship you describe cannot be true in general, since you
> > appear not to have imposed any constraints on the sigmas.
> > If sigma_Max * sigma_Min > 1, you would have rho imaginary.
> > And you would have rho = 1 (or -1, you can't tell which)
> > iff sigma*Min = 0, which is unreasonable.
------------------------------------------------------------------------
Donald F. Burrill [EMAIL PROTECTED]
348 Hyde Hall, Plymouth State College, [EMAIL PROTECTED]
MSC #29, Plymouth, NH 03264 603-535-2597
184 Nashua Road, Bedford, NH 03110 603-471-7128
