Law of the large numbers

Given a random variable |$X$|⁠, its expected value and variance are given by |$E[X]\,{=} \langle x \rangle \,{=}\, \sum _i P(x_i)x_i$| and |$Var(x) \,{=}\, \sigma^2(x) \,{=}\, E[(x-\langle x \rangle)^2] \,{=}\, \sum_iP(x_i)(x_i-\langle x \rangle )^2$|⁠. Tchebyshev’s lemma states that the probability of the occurrence of |$\vert x\,{-}\,\langle x \rangle \vert \,{\gt}\, t$| is less than or equal to |$\sigma^2(x)/t^2$|⁠. The proof is as follows: and thus it holds: which is the statement of Tchebyshev’s lemma. Let us consider now the law of large numbers. Be and the variables |$X_i$| are mutually independent and all distributed according to the same probability distribution. Let finally, let |$\langle x \rangle$| and |$\sigma^2 (x)$| be finite. Let us notice that, thanks to the independence of the variables, we have that the expected value of |$NY_N$| is equal to |$N \langle x \rangle $|⁠, and its variance is equal to |$N \sigma^2(x)$|⁠. The law of large numbers states that the probability that |$\vert Y_N-\langle x \rangle \vert$| is greater than |$epsilon$| tends to |$0$| when |$N \,{\rightarrow}\, \infty$|⁠. The proof is an immediate consequence of Tchebyshev’s lemma. This tells us that: and, choosing |$t \,{=}\, N \epsilon$|⁠, one has: with the quantity on the right-hand side of Eq. (A.5) that tends to zero when |$N$| tends to |$\infty$|⁠. We have, therefore, proved the law of large numbers.