I have two different independent, normally distributed random samples that have the same parameters. $X_1$ ... $X_n$ is distributed $N(\mu_1,\sigma_1)$ and $Y_1$ ... $Y_n$ is distributed $N(\mu_2, \sigma_2)$. If $\mu_1=\mu_2=\mu$ and $\sigma_1=\sigma_2=\sigma$, the MLE of these would just be derived the same as any normal random distribution, right?
I'm assuming yes because if $\mu_1=\mu_2$ and $\sigma_1=\sigma_2$, both of these random variables (and the technical "third" rv) should be the same. I just wanted confirmation that I'm correct before doing all the work and finding out that this isn't the right approach.
I don't want anyone to calculate the MLE for me. I just want to make sure I'm right. Thank you!