Re: uncorrelated pair of rvs in the bivariate normal distribution

On Mar 26, 3:12 pm, Paul Rubin <ru...@xxxxxxx> wrote:
lydiajone...@xxxxxxx wrote:
On Mar 26, 11:43 am, lydiajone...@xxxxxxx wrote:
On Mar 26, 11:26 am, Paul Rubin <ru...@xxxxxxx> wrote:

lydiajone...@xxxxxxx wrote:
Dear forumers,
Let (X,Y) be distributed according to a bivariate normal distribution
with means (a,b) and covariance structure (sigma^2,tau^2,rho)
I would like to show that Xbar and (Y-b)^2 are uncorrelated
I guess I should work on E(Xbar)(Y-b)^2, but I do not know where to go
from there. I multiplied out (Xbar)(Y-b)^2 and I get several terms
which do not seem to cancel out. One of the terms I get is the raw
moment muXY^2.
Thanks a lot for the help.
I think you need to clarify a couple of things here. Is Xbar a sample
mean? If so, (a) is the sample i.i.d. and (b) do you have n
observations of X but only one observation of Y?
/Paul
Thanks for the message. The sample is n iid pairs of observations
(X1,Y1) ...(Xn,Yn).
I have been searching the literature for a reference on the moments of
the bivariate normal distribution, but I have not found much.

And I forgot to mention, Xbar is the empirical mean. Thanks.

But are you squaring Y-b for some arbitrary instance of Y, rather than
for a sample point (or squaring Ybar - b)? If you obtain a sample (Xi,
Yi), compute Xbar from the sample, then obtain an independent new
observation of Y, the new Y would be independent of Xbar and thus
(trivially) uncorrelated.


By Y, I meant Yi.

Are you perhaps looking at Cor(Xbar, (Yi-b)^2) for an arbitrary index i
(i.e., Y is one of the sample Yi)?

/Paul

Since the Yis are identically distributed, I think that writing Y is
just fine.
.

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