I'm just confused about the purpose of MLE.
What I've read about MLE for normal distribution is that it is a method that will find the values of parameters (μ and σ in this case) that result in the curve that best fits the data, or in simpler version maximize the probability of observing our data.
This makes sense when it comes to normal distribution, but I can't imagine a best 'curve' for Bernoulli distribution, what is the point of having MLE in this case?
The likelihood is a function of the parameter, considering $\mathbf{x}$ as given data. Thus for bernulli distribution
$$L(\theta)=\theta^k(1-\theta)^{n-k}$$
Where $k=\Sigma_i X_i$ and $\theta \in (0;1)$
To maximize L it is easy use logL
$$logL=klog\theta+(n-k)log(1-\theta)$$
Derivating in $\theta$ and setting =0 you get
$$\hat{\theta}=\frac{k}{n}=\overline {X}_n$$