
MLE of bivariate normal
how to calculate the MLEs of the 5 parameters of a bivariate normal population, the parameters being $\displaystyle \mu_x,\mu_y,\sigma_x,\sigma_y,\rho$?
while trying to solve for $\displaystyle \mu_x$, i got the following equation
$\displaystyle \sum{\frac{(x\mu_x)}{\sigma_x^2}} = \rho\sum{\frac{(y\mu_y)}{\sigma_x\sigma_y}}$
is this correct? how can i solve for $\displaystyle \mu_x$ from here?

you need to take the partial derivative wrt to all 5 parameters
It's smart to take the logarithm of the joint density first

I took the logarithm and differentiate partially wrt $\displaystyle \mu$ and got that

you need to solve all five together, the MLE of $\displaystyle \mu_X$ should be $\displaystyle \bar X$
and those for the population variances should be the sample variances with n in the denomiator.