[Math] Symmetric property for bivariate normal distribution

Question

I'm trying to prove that the bivariate normal distribution has the symmetric property.
I.E. N2(a,b;p)=N2(b,a;p) where a, b are constants (and the upper bound for their respective integrals.) and p is the correlation. Also assuming that x1 and x2 are standard normal (N(0,1)).

So far, I've tried using a conditioning approach and attempting to integrate, any help would be greatly appreciated!

Thanks!

Answer 1

To sum up what transpires from the question and from some comments, it seems that the question is as follows:

Let $(X_1,X_2)$ denote a centered gaussian vector with covariance matrix $C=\begin{pmatrix}1 & p\\ p & 1\end{pmatrix}$. Then,$$ \mathbb P(X_1\leqslant a,X_2\leqslant b)=\mathbb P(X_1\leqslant b,X_2\leqslant a). $$

To show this, note that the matrix $C$ stays the same when one exchanges the columns and one exchanges the rows, hence $(X_2,X_1)$ is distributed like $(X_1,X_2)$ (this uses the fact that the distribution of a gaussian vector is entirely determined by the vector of its means and by its covariance matrix). In particular, $$ \mathbb P((X_2,X_1)\in(-\infty,a]\times(-\infty,b])=\mathbb P((X_1,X_2)\in(-\infty,a]\times(-\infty,b]), $$ which is the desired result.