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 $\hat{\mu}_n=\bar{X}_n=\frac{1}{n}\sum_{i=1}^nx_i$

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    $\begingroup$ Because the random variables are independent, you can find the values of $\mu_1$ and $\mu_2$ that maximize the product of the probability density functions or to make the calculations simpler maximize the sum of the logs of the probability density functions. In other words, "maximize the likelihood". $\endgroup$ – JimB Mar 26 at 3:43
  • $\begingroup$ First step is to write down the likelihood function which is just the pdf of $(X_1,X_2,X_3)$. Then proceed as usual. $\endgroup$ – StubbornAtom Mar 26 at 6:45

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