Hello! If someone can help me, I would really appreciatte!

Suppose we have a mixture of normals

$\displaystyle f(x; \mu_1 , \mu_2, \sigma, p) = pN(\mu_1, \sigma^2) + (1 - p)N(\mu_2, \sigma^2). $

My problem is to compute the MLE's. For example, I got the same MLE for $\displaystyle \mu_1 $ and $\displaystyle \mu_2$, the sample mean. But this result doesn't make much sense for me... because if it is correct, the estimates are the same, the normals distributions are equal and the likelihood, which was a mixture of normals, becomes a single normal distribution, independently of p.

If I have a histogram from the original distribution and the modes are really far apart, and also if the first mode is way smaller than the second, for example, the MLE's obtained before are not going to estimate it correctly, since it is considering p = 1/2 and the same mean.

What am I doing wrong? Can someone help me here, please?