According to wikipedia, the maximum likelihood estimator for the population variance, $\sigma^2$ is $$ s^2 = \frac{1}{n}\sum_i^n (x_i - \bar{x})^2 $$
where $\bar{x}$ is the mean of the samples.
I'm not well versed with MLE, but I just watched a video https://www.youtube.com/watch?v=iDezY7m1S3g&ab_channel=BenLambert of how the maximum likelihood estimator for the population mean and variance of data sampled from a gaussian distribution can be derived, and it resulted in the above formula for the variance. This seems only possible because the gaussian distribution has the population mean and variance in the pdf.
How would you perform MLE on a pdf that doesn't have the mean and variance in the pdf itself? e.g., how would you do it for the uniform distribution?