# Thread: bivariate normal distribution...

1. ## bivariate normal distribution...

if (X,Y) have bivariate normal distribution N_2(0,S), S = (s_ij)
s_11 = s_22 = 1 and s_12 = s_21 = p (correlation coefficient)

is it correct to say that X,Y are both normal with N(0,1)?

furthermore, is it right to say that since p = cov(X,Y) / (s_1 * s _2) <=>

p = cov(X,Y) = E(XY) + 0 + 0 + 0 = E(x^2) = 1

???

2. Originally Posted by sezmin
if (X,Y) have bivariate normal distribution N_2(0,S), S = (s_ij)
s_11 = s_22 = 1 and s_12 = s_21 = p (correlation coefficient)

is it correct to say that X,Y are both normal with N(0,1)?
Yes

furthermore, is it right to say that since p = cov(X,Y) / (s_1 * s _2) <=>

p = cov(X,Y) = E(XY) + 0 + 0 + 0 = E(x^2) = 1

???
No ! Why would E[XY]=E[X^2] ??
Let's come to think about that : if it was possible to compute the covariance, they wouldn't have put p. You just can't compute it with the information you're given.

3. Ah, ok. Thanks!

Then...if I had an arbitrary function g(t) defined on [0, infinity)

E[ h(X^2) * (Y^2) ] =/= E[ h(X^2) * (X^2) ]

correct?

4. Yes, it sure is different ! (unless you have very particular conditions)
The correlations of X and X, and of X and Y are not always the same, in particular !

5. The correlation of X and X is 1.