I have three questions: What is the interpretation of the outcome of the Probability Density Function (PDF) at a particular point? How is this result related to probability? What we exactly do when we maximize the likelihood?
To better explain my questions:
(i) Consider a continuous random variable X with a normal distribution such that and . If we evaluate the PDF at a particular point, say 3.4, using the formula:
we get . How we interpret this value?
(ii) I previously read that the result of 0.1144 is not necessarily the probability that X takes the value of 3.4. But how the result is related to probability concept?
(iii) Consider a sample of the continuous random variable X of size N=2.5, such that and . We can use this sample to maximize the log-likelihood:
If f(X) is not exactly a probability, what are we maximizing? Some texts detail that "we are maximizing the probability that a model (set of parameters) reproduces the original data". Is this phrase incorrect?