I am working through find the maximum likelihood estimators of the bivariate normal distribution, without using matrices. I have the following density function:

$f(Y_1,Y_2) = \frac{1}{2\pi\sigma_1\sigma_2\sqrt{1-\rho_{12}^2}} \exp \bigg\{ -\frac{1}{2(1-\rho_{12}^2)} \bigg[ \bigg(\frac{Y_1 - \mu_1}{\sigma_1} \bigg)^2 -2\rho_{12} \bigg( \frac{Y_1 - \mu_1}{\sigma_1} \bigg)\bigg( \frac{Y_2 - \mu_2}{\sigma_2} \bigg) + \bigg( \frac{Y_2 - \mu_2}{\sigma_2} \bigg)^2 \bigg] \bigg\}$

So far, I've obtained the log likelihood:

\begin{equation} \begin{split} \ell(\mu_1, \mu_2, \sigma_1, \sigma_2,\rho_{12}) & = -n\log2 \pi - n\log \sigma_1 - n\log \sigma_2 - \frac{n}{2} \log(1-\rho_{12}^2) \\&- \frac{1}{2(1-\rho_{12}^2)}\sum_{i=1}^n \bigg[ \bigg(\frac{Y_1 - \mu_1}{\sigma_1} \bigg)^2 -2\rho_{12} \bigg( \frac{Y_1 - \mu_1}{\sigma_1} \bigg)\bigg( \frac{Y_2 - \mu_2}{\sigma_2} \bigg) + \bigg( \frac{Y_2 - \mu_2}{\sigma_2} \bigg)^2 \bigg] \end{split} \end{equation}

I'm having trouble with next steps: taking the partial derivatives w.r.t. each parameter, setting them equal to 0, and solving for the parameters.

Is there a better way to write the log likelihood so that I can more easily take each partial derivative?


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