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?

Thanks! Please feel free to add more background to your answer..

  • 1
    $\begingroup$ Could you indicate which of these concepts is not yet explained at stats.stackexchange.com/questions/50? $\endgroup$
    – whuber
    Feb 28, 2020 at 0:48
  • $\begingroup$ @whuber I think I understand the idea of a random variable now from your post and where the randomness comes from: To confirm: - A random variable is just a map, or an assignment, of elements from your sample space to a number (which is often their numerical implication) I'm not sure yet whether I understand the position and relationship between population, the "underlying probability distribution" and the "model" in this context. Could you please expand on this? - And also, I'm not sure whether I'm clear on where the "randomness" comes from - could you articulate that? $\endgroup$
    – user523384
    Feb 28, 2020 at 1:04
  • $\begingroup$ The entire description of the tickets in the box, their quantities, and what is written on them is the model. "Population" is an ill-conceived term, applicable only in limited circumstances with many slightly differing interpretations, and so is best avoided; but where it is applicable, you may think of it as being the set of the individual tickets. The probability differs from a probability distribution: the former describes how many of each type of ticket is in the box while the latter depends on the random variable. The randomness comes from mixing the tickets and drawing one. $\endgroup$
    – whuber
    Feb 28, 2020 at 14:06
  • 2
    $\begingroup$ I agree with @whuber , and might suggest in addition that people use the expression "data generating process" (DGP) in place of "population" to clarify and reduce the confusion. After all, the sampled data target the process that produced those data. In nearly all cases, even in ostensibly "population sampling" cases, this process differs in various important ways from the "strict population/ball and urn" framework of textbooks. (There are numerous sources of bias which nearly always exist). On the other hand, the "underlying distribution" and "model" are both meant to mimic the real DGP. $\endgroup$ Feb 28, 2020 at 14:37


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