# What is the difference between a "population," a "sample space," an "underlying probability distribution? and a "model"?

I'm trying to understand an overview of the topic of statistical inference. I have learnt bits and pieces of many of the probability and statistics involved in it but before learning it rigorously it occurred to me that I should really have a good basic foundation of what is going on overall.

On Wikipedia, the subject "Statistical inference" has the following definition:

Statistical inference is the process of using data analysis to deduce properties of an underlying probability distribution. Inferential statistical analysis infers properties of a population, for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is sampled from a larger population....Given a hypothesis about a population, for which we wish to draw inferences, statistical inference consists of (first) selecting a statistical model of the process that generates the data and (second) deducing propositions from the model.

I wanted to understand these ideas a little clearer: What is the relationship between a "population" and an "underlying probability distribution" - it seems that they are used interchangeably. And what is the relationship with these topics with random variables and their distributions which we often jump into when modelling?

For example: If we have a population, and we're considering the height of the people in the population. Would you say that:

• The population is the sample space containing all the possible people (say $$\omega_1, \omega_2,....$$)?
• The height of a certain person in the population as $$X(\omega_i)$$? (Why would this be random? - Or is it saying that "If I select one of the people at random $$\omega$$, then $$X(\omega)$$ would be the "random" height?
• The underlying probability distribution - Would this be the distribution of $$X$$? (If so, does this mean that deducing properties for this distribution is deducing properties about a specific characteristic (i.e. the height) of the population (and hence about the population in general?)) Additional Q: If this is true, then why are we interested in the distribution of the sample data? Weren't we interested in deducing properties of the population, not the data?
• The selection of statistical model: Is that our assumption about the functional form for the distribution of $$X$$? (whether that be the PDF or the PMF) What is the "process that generates the data" in this context that we are trying to model?