What exactly f(x;θ) means?

[lsiqueirap]

So, almost all books use the notation f(x;θ) when explaining statistical models. For example:

Unfortunatelly, it isn't well explained what really is f (x; θ). Is it the distribution function of the population from which my observations (X_1, ..., X_n) were made? What is the role of theta in this notation, is it there just to point out that this is the distribution function of the population whose parameter we are trying to estimate?

 

[Romsek]

theta is considered a parameter of the density function while x is considered it's variable.

Consider the exponential distribution

$\displaystyle p_{\theta}(x) = \theta e^{-\theta x}$

or say a symmetric uniform distribution

$\displaystyle p_{\theta}(x) = \dfrac{1}{2\theta},~x \in [-\theta,\theta]$

Do you see how theta acts as a parameter?

 

[lsiqueirap]

theta is considered a parameter of the density function while x is considered it's variable.

Consider the exponential distribution

$\displaystyle p_{\theta}(x) = \theta e^{-\theta x}$

or say a symmetric uniform distribution

$\displaystyle p_{\theta}(x) = \dfrac{1}{2\theta},~x \in [-\theta,\theta]$

Do you see how theta acts as a parameter?

Yes, I do, it makes sense. But, the first question stills: is this function the density/mass function of the population from which my observations were made? In positive case, how would I know if it's a poison, binomial or any other if, in practice, I don't know the entire population?

 

[Romsek]

Yes f is the distribution of some population. Without more information you don't know a priori what the form of the density is but in problems generally you are told the density and may or may not be given a value for the parameter.

It's a common problem given a set of samples from some population, and a parameterized density that describes it, to come up with a "best" in some sense estimate of the parameter.