How does a Random Sample relate to Random Processes and Random Variables?

Question

What is the difference between a Random Sample, Random Variable (RV) and Random Process (RP)? As far as I know, a RV is a mapping from an experimental space to the real numbers and a RP is a mapping from an experimental space to a set of deterministic functions each with an associated probability. If we sample these functions at a particular instance then we get a RV from the sample of all these deterministic functions at that instance. Now where does a Random Sample come in to this. As far as I know the statistics from a Random Sample or a Sampling Distribution are also treated as Random Variables, for example the Sample Mean ($\bar{X}$) so in this case if our Sampling Distribution is a Random Variable then is the Random Sample an observation of it and a sort of sample of it just like a RV is for a RP?

I have a decent background in Probability theory but I am new to statistics, in general. So I just wanted to clear up some fundamental concepts before I dive into the advanced stuff.

Answer 1

An example of a random sample: $n$ i.i.d. draws $X_1, \ldots, X_n$ from a distribution $P$. Each $X_i$ is a random variable. You can also view $(X_1, \ldots, X_n)$ as a single draw from the product distribution $P \times \cdots \times P$ (which could be interpreted as a random process on $\{1, \ldots, n\}$ although this is not really helpful). Statistics like the sample mean $\bar{X}$ are functions of the sample $X_1, \ldots, X_n$, and are therefore also random variables.

Loosely speaking, one common task in statistics is using observations $X_1, \ldots, X_n$ to make inferences about the unknown underlying distribution $P$.