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.

Answer 2

The bivariate normal distribution is conventionally parametrized as follows: $$ N_2\left(\begin{bmatrix} \mu_1 \\ \mu_2 \end{bmatrix}, \begin{bmatrix} \sigma^2 & \rho\sigma\tau \\ \rho\sigma\tau & \tau^2 \end{bmatrix}\right) $$ where the $2\times1$ matrix is the expected value and the $2\times 2$ matrix is the variance, also called the covariance matrix because its entries are the covariances.

What your notation means I've decided not to try to guess.

Later edit informed by a comment below: OK, so you intended $N_2$ to be a cumulative distribution function. If you're trying to integrate something, you're certainly making the problem more complicated than it is. Everything about the definition of the distribution treats the two variables symmetrically.