Sorry for another question on MLE and PDF.
I understand ( or at least I think I do) how MLE is used to find the parameters for a given distribution such as a Gaussian one but I have a question based on the definition of MLE and PDF.
Let’s supposed that I’m collecting data for an experiment which is known to be a continuous Gaussian and I collect the following numbers : 10,10,11,12,13,12
I know how to use these numbers to calculate the mean and variance for the distribution.
Per my understanding, a more formal definition would be : I’m finding the mean and variance that maximize likelihood of the joint probability P(10) x P(10) x P (11) x P(12)..
Since my variable is continuous, the probability of a given number is always zero by definition. So, P(10)=0, P(11) = 0, etc.
What am I missing here?
The probability of getting exactly those values from a continuous distribution would indeed be $0$. The likelihood you are maximizing is not a probability, but a probability density.