### Concours Centrale Supelec: mathématiques 2 MP

Quite an interesting, coherent problem, combining analysis and probability.

There were some pretty tricky sections:

- Questions 22 to 25, which demonstrated the uniform convergence of a sequence of functions to the density of the normal distribution \(\mathcal{N}(0,1)\), \(\varphi(t)= \frac{1}{\sqrt{2\pi}}e^{-\frac{t^2}{2}}\)
- Question 31 demonstrated a special case of the central limit theorem. \[\mathrm{P}(\frac{\sum_{i=1}^n X_i}{\sqrt{n}}\geqslant u )\underset{n\to+\infty}{\rightarrow}\int_u^{+\infty} \varphi(t)\mathrm{d} t\] The \(X_i\) are here independent VA which take the value -1 or 1 with probability \(\frac{1}{2}\).
- Question 33 was an application to demonstrate a tension criterion.

We could observe that the normal distribution is a limit of a sequence of binomial distributions, or that the binomial distribution is a kind of discretization of the normal distribution.

In fact, the central limit theorem is much more powerful, since the result demonstrated in question 31 extends to any sequence of independent, centered ILs with the same distribution and variance \(\sigma^2_X\). We say that the VA \[\frac{\sum_{i=1}^n X_i}{\sqrt{n}}\] converges in law to the normal law \(\mathcal{N}(0,\sigma^2_X)\).

Answers to the Maths 2 test on May 12 in the MP Centrale Supelec 2023 exam:

Problem set: