if (x y) ~ bivariate normal (0, 0, 1, 1, ρ), show that q = (x^2 −2ρxy+y^2)/ 1−ρ^2 is distributed as chi square (2 degrees of freedom).

Question

Here X and Y follow bivariate normal distribution and Q is a new variable including X and Y. Prove that the new variable Q follows chi square distribution with 2 degree of freedom.

Answer 1

If (x y) is a k-dimensional Gaussian random vector with mean vector $\mu$ and rank k covariance matrix C, then q=((x y) -$\mu^T$)$C^{-1}$((x y)$^T$-$\mu$) is chi squared distributed with k degrees of freedom. (https://en.wikipedia.org/wiki/Chi-squared_distribution)

Here is another interesting link explaining the relationship between multivariate normal distribution and chi squared distribution: https://statproofbook.github.io/P/mvn-chi2.html