I am trying to grab on to some intuition about the area where random variables start looking a bit more like calculus. I've learned about random variables and the weak law of large numbers, but seem to not get the idea of "sequences of random variables" and their behavior.

the issue came up following a question my teacher gave me (attached below). I seem to struggle at the very beginning of the question, as i cannot fully organize the data and understand the question before even beginning to solve it.

I would really appreciate some guidance on how to approach such questions and how to make more sense out of them.

The observations I made were:

  • It seems to have something similar with throwing dices
  • The convergence to $\frac{1}{3.5}$ is similar to $\frac{1}{E_{dice}(X)}$

things that are not clear enough for me are:

  • what $S_n$ actually represents
  • what $Y_m$ actually represents

The original question

Let $ \{X_i\}_{i=1}^{\infty} $ be a series of equally distributed variables, and $S_n = \sum_{i=1}^{n}{X_i}$

Let us define $k$ as a "known" number, such that $\exists n .S_n=k$

For every $m\in\mathbb{N}$ Let us define $T_m$ as the number of "known" numbers which are not bigger than $m$, and $Y_m=\frac{T_m}{m}$

Assuming $ \{X_i\}_{i=1}^{\infty} $ is a series of independent variables and $X_1\sim U[1,6]$ (discrete uniform),

Prove that:

$\lim_{m\to \infty}P(|Y_m-\frac{1}{3.5}|>0.001)=0$


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