Confusion about random variables and convergence in probabilty and distribution

manpreet Best Answer 2 years ago

I'm studying statistical analysis and there's something fundamental I'm missing about random variables and how they are used in defining convergence in probability or distribution:

In my syllabus (which is in dutch, so the terms i use might be slightly off), when talking about samples, it says that

If we want to study the properties of a random variable XX in a target population, we take a random aselect sample of nn subjects from a collection of nn random variables X1,...XnX1,...Xn that are all mutually (pairwise?) independent and which all have the same distribution, namely that of XX in the target population.

A bit further, discussing convergence, it says

An infinite row of random variables X1,X2,...X1,X2,... on a probability space converges in probability to XX if the folowing is true for each ϵ>0ϵ>0: limn→∞P(|Xn−X|)≥0limn→∞P(|Xn−X|)≥0

What I don't understand is what XiXi actually means in these two contexts. I read it as follows: In the first part, it is presented as one choice from the population: XiXi is the length of the