I have to calculate the MLE of the independent random variables $$X_1\sim N(\mu_1,1),X_2\sim N(\mu_2,1),X_3\sim N(\mu_1+\mu_2,2)$$, where $N$ is the normal distribution, how do I do this?
So far I learned to calculate the MLE for one dimensional parameters, and same-distributed random variables. For example, if a have a random sample $$\{X_i\}_{i=1}^n\overset{iid}{\sim}N(\mu,\sigma_0^2)$$, where $\sigma_0^2$ is a known parameter, then, the MLE is $\hat{\mu}_n=\bar{X}_n=\frac{1}{n}\sum_{i=1}^nx_i$