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I'm a bachelor's student, and I'm quite frustrated by how the probability courses in my university are set up. I feel like we go straight into complex things without adequately covering the basics.

One of the most common examples for this is independence. The only definitions of independence we saw were mutual independence of random variables and pairwise independence. I know that there also exist "intermediate" notions of independence, ie we can say $(X_1,...,X_n)$ and $(X_{n+1},...,X_{n+m})$ are independent with an analogous definition. Frequently I encounter situations where the $X_i$ are all pairwise independent, and apparently we are supposed to conclude that $$X_1+...+X_n \text{ and } X_{n+1}+...+X_{n+m}$$ are two independent random variables. Is this sort of reasoning true? I would also appreciate any clarifications on how independence works and is shown in general, but I am aware that that would be a very vague question and that I will probably just need to take the time to read a book and figure it out after courses are out.

ThighCrush
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  • It isn't clear what you don't understand. Do you understand why the outcomes of a coin toss and die roll are independent? To show independence we usually show something like $P(A|B)=P(A)$ or $P(A\text{ and }B)=P(A)P(B)$. – John Douma Jun 22 '22 at 17:18
  • Please edit your post for clarity. As you are aware, defining a notion is very different from proving all manner of theorem regarding it. Start with the definition, then look at many examples, then look at some of the theorems. – lulu Jun 22 '22 at 17:36

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