In 'classic' math when you have a function like $\sin (\theta)$ or $\cos (\pi)$ is pretty straight forward that both should have an argument and it's very simple to see that in this specific case the first would be $\theta$ and the second one would be $\pi$ . But when dealing with statistic related functions - mainly p.d.f's - it is getting very messy to me. It's confusing when what's inside of the parenthesis it's an argument, a parameter, a conditional probability.
It was all clear last semester that $\theta$ is normally conventioned as a parameter representing the mean, or maybe $\sigma$ representing the standard deviation. But now - my current subject is the maximum likelihood and other parameter estimation methods - suddenly $\theta$ is getting used everywhere and i don't know if it's a general parameter, it's still the mean, it's the argument of the p.d.f. ...
I found that a parameter, $\theta$, is a function of the probability density function, but that confused me even more. So what's the argument vs. parameter relation? Are the parameters countable like you have the standard deviation, the mean and the variance or are they uncountable, generated each time for a specific purpose? Thanks!
Last semester, $\theta$ was the (collection of) parameter(s) for a model or distribution. You were expected to deduce things about various statistics of the distribution or of samples from the distribution.
In maximum likelihood estimation, instead of going from a p.d.f. to a sample, one has a sample and a model and would like to know what model parameters best produce the sample. So it's running the same problems in the reverse direction. So this semester, $\theta$ is still the (collection of) parameter(s) for a model or distribution. You are expected to deduce the parameters from the model(s) and sample(s). $\theta$ is not the argument to the p.d.f.
The question of whether parameters are countable or uncountable is really a question about model selection. If you select a model with only a few "knobs", like the normal distribution, then the set of parameters is finite (and countable). If you don't select a model at all, then there are an infinite number of "knobs" (to set the p.d.f. or p.m.f. at each point of the support) and the set of parameters is infinite (and either countable (p.m.f.) or uncountable (p.d.f.)).