Suppose we observe $Y_i\sim \mathcal{N}(\theta_0 + \theta_1 x_i, \sigma_i^2)$, with $x_i$ and $\sigma_i^2$ known for all $i = 1,\ldots,n$ and $Y_1,\ldots,Y_n$ independent. Assume $\theta_0$ is unknown and $\overline{x}=0.$
What is the MLE of $\theta_1$? The fact that the variances are different is throwing me off. I end up getting that I should maximize $\exp{\frac{\sum_i 2y_i(\theta_0 + \theta_1 x_i) - (\theta_0+\theta_1 x_i)^2}{2\sigma_i^2}}.$ From there I'm stuck because taking partial derivatives doesn't give me anything.
Thanks!