Statistical Indep and Corr of Normal RVs

Hi,

I have come across the fact that for 2 normal rvs, x and y, Corr(x,y) =
0 does not imply x and y are independent. It seems to me however that
this contradicts the condition that x and y are independent iff their
pdfs separate. Here is the simple argument I am thinking of:

In a problem I am looking at, x and y are two normal rvs both with mean
zero and sdev=1. In this case, their joint pdf is given by the standard
bivariate normal pdf:

f(x,y) = 1/[sqrt(1-r^2)*2*pi] * exp[ - { x^2 -2*x*y*r +
y^2}/(2*(1-r^2))]

But r=0 implies this just reduces down to
f(x,y | r=0) = 1/[2*pi] * exp[ - { x^2 + y^2}/2]

which can be written as the individual pdfs for x and y.

Since stat independence is also equivalent to being able to write the
joint pdfs of the two rvs and the product of their individual pdfs,
does zero correlation in this case imply stat indep? If this is the
case, what happens in those cases where r=0 but x and y are not indep
that causes this argument to break down?

TIA,

Matt

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