$1$-Objectives :

We provide for every student a dataplay. i.e., a vector $(X_1;…;X_{500})inmathbb{R}^{500}$.

This vector is the realization of a 500-sample $(X_1;…;X_{500})$ of a random variable X whose law of probability is unknown.We denote by f the probability density of X with respect to the Lebesgue measure on $mathbb{R}$ and F its distribution function.

The purpose of this project is twofold. On the one hand, it is a question of becoming familiar with the software R, and on the other hand, of finding the probability law of X from the observations $(x_1;…; x_{500})$.

We first assume that the probability law of X belongs to a set of classical probability laws whose list is given below with their codes on R.

$2$– Questions :

All the classical laws proposed above are parametric. In other words, f depends on a parameter $theta$. Exemple pour la loi de Pareto: $theta$ =(m,s), $f_{Pareto}(x;θ) = frac{s}{m(s−1)}(1+frac{x}{m(s−1)})^{−s−1}$.

Question 1: Décrire vos données par une simple statistique descriptive (on résumera

les résultats dans un seul tableau). Commentezanswer 1:enter image description here

Question 2 : Let ${Y_{k}}_{kgeq 1}$ be the sequence defined by $Y_k = X_{L(k)}$ with $L(1) = 1$ ; L(k + 1) = min{$j : j > L(k); X_{j} > X_{L(k)}$}; $kgeq1:$

So define the sequences of the v.a.r ${I_{k}}_{kgeq1}$ et ${N_{n}}_{ngeq1}$ with $I_1=1$ p.s

$I_k=mathbb{1}_{{X_k > max(X_1;…;X_{k-1})}}, kgeq 2; et N_n=sum_{k=1}^{n}I_k$.Q2.1 : Propose a code on R which calculates respectively ${L_{k}} , {Y_{k}}$ et $N_n$ for a

sample given $(X_1;…;X_{n})$. Apply this code to your data.

answer 2.1 : enter image description here enter image description here

Q2.2 : Show that ${I_{k}}$ is a sequence of independent random variables et ${I_{k}}simmathbb{B}er(1/k)$.

Q2.3 : Justify why, when $ntoinfty$ , $N_n /log{n}xrightarrow{p.s} 1 $ et $(N_n-log{n})/sqrt{log{n}}xrightarrow{mathcal{L}}mathcal{N}(0,1)$