Bivariate normal distribution

For means ${\mu }_i$, standard deviations ${\sigma }_i$, and correlation $\rho$, the standardized quadratic form in the density is

A bivariate standard normal distribution is a bivariate normal distribution whose two marginal means are zero and marginal variances are one; its remaining parameter is their correlation.

For independent $Z_i\sim N_p(0,\Sigma )$ with known zero mean,

Writing $a={\Sigma }_{11}$, $c={\Sigma }_{12}$, and $b={\Sigma }_{22}$, the vector $({\hat{\Sigma }}_{11},{\hat{\Sigma }}_{12},{\hat{\Sigma }}_{22})$ has asymptotic covariance

Conditionally on $X_2=x_2$,