grandes-ecoles

Papers (191)
2025
centrale-maths1__official 40 centrale-maths2__official 42 mines-ponts-maths1__mp 20 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 24 mines-ponts-maths2__psi 26 polytechnique-maths-a__mp 27 polytechnique-maths__fui 16 polytechnique-maths__pc 27 x-ens-maths-a__mp 18 x-ens-maths-c__mp 9 x-ens-maths-d__mp 38 x-ens-maths__pc 27 x-ens-maths__psi 38
2024
centrale-maths1__official 28 centrale-maths2__official 29 geipi-polytech__maths 9 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 19 mines-ponts-maths2__mp 23 mines-ponts-maths2__pc 21 mines-ponts-maths2__psi 21 polytechnique-maths-a__mp 44 polytechnique-maths-b__mp 37 x-ens-maths-a__mp 43 x-ens-maths-b__mp 35 x-ens-maths-c__mp 22 x-ens-maths-d__mp 45 x-ens-maths__pc 24 x-ens-maths__psi 26
2023
centrale-maths1__official 44 centrale-maths2__official 33 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 15 mines-ponts-maths1__pc 23 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 18 mines-ponts-maths2__psi 22 polytechnique-maths__fui 23 x-ens-maths-a__mp 25 x-ens-maths-b__mp 24 x-ens-maths-c__mp 20 x-ens-maths-d__mp 20 x-ens-maths__pc 18 x-ens-maths__psi 15
2022
centrale-maths1__mp 48 centrale-maths1__official 48 centrale-maths1__pc 37 centrale-maths1__psi 43 centrale-maths2__mp 32 centrale-maths2__official 32 centrale-maths2__pc 39 centrale-maths2__psi 45 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 24 mines-ponts-maths1__psi 24 mines-ponts-maths2__mp 24 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 x-ens-maths-a__mp 13 x-ens-maths-b__mp 40 x-ens-maths-c__mp 27 x-ens-maths-d__mp 46 x-ens-maths1__mp 13 x-ens-maths2__mp 40 x-ens-maths__pc 15 x-ens-maths__pc_cpge 15 x-ens-maths__psi 22 x-ens-maths__psi_cpge 23
2021
centrale-maths1__mp 40 centrale-maths1__official 40 centrale-maths1__pc 36 centrale-maths1__psi 29 centrale-maths2__mp 30 centrale-maths2__official 29 centrale-maths2__pc 38 centrale-maths2__psi 37 x-ens-maths2__mp 39 x-ens-maths__pc 44
2020
centrale-maths1__mp 42 centrale-maths1__official 42 centrale-maths1__pc 36 centrale-maths1__psi 40 centrale-maths2__mp 38 centrale-maths2__official 38 centrale-maths2__pc 40 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 24 mines-ponts-maths2__mp_cpge 21 x-ens-maths-a__mp_cpge 18 x-ens-maths-b__mp_cpge 20 x-ens-maths-d__mp 14 x-ens-maths1__mp 18 x-ens-maths2__mp 20 x-ens-maths__pc 18
2019
centrale-maths1__mp 37 centrale-maths1__official 37 centrale-maths1__pc 40 centrale-maths1__psi 39 centrale-maths2__mp 37 centrale-maths2__official 37 centrale-maths2__pc 39 centrale-maths2__psi 49 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 26
2018
centrale-maths1__mp 47 centrale-maths1__official 47 centrale-maths1__pc 41 centrale-maths1__psi 44 centrale-maths2__mp 44 centrale-maths2__official 44 centrale-maths2__pc 35 centrale-maths2__psi 38 x-ens-maths1__mp 19 x-ens-maths2__mp 17 x-ens-maths__pc 22 x-ens-maths__psi 24
2017
centrale-maths1__mp 45 centrale-maths1__official 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__official 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 26 x-ens-maths2__mp 16 x-ens-maths__pc 18 x-ens-maths__psi 26
2016
centrale-maths1__mp 42 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 47 centrale-maths2__psi 27 x-ens-maths1__mp 18 x-ens-maths2__mp 46 x-ens-maths__pc 15 x-ens-maths__psi 20
2015
centrale-maths1__mp 42 centrale-maths1__pc 18 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 18 centrale-maths2__psi 33 x-ens-maths1__mp 16 x-ens-maths2__mp 31 x-ens-maths__pc 30 x-ens-maths__psi 22
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 27 centrale-maths2__mp 24 centrale-maths2__pc 26 centrale-maths2__psi 27 x-ens-maths1__mp 9 x-ens-maths2__mp 16 x-ens-maths__pc 4 x-ens-maths__psi 24
2013
centrale-maths1__mp 22 centrale-maths1__pc 45 centrale-maths1__psi 29 centrale-maths2__mp 31 centrale-maths2__pc 52 centrale-maths2__psi 32 x-ens-maths1__mp 24 x-ens-maths2__mp 35 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__mp 36 centrale-maths1__pc 28 centrale-maths1__psi 33 centrale-maths2__mp 27 centrale-maths2__psi 18
2011
centrale-maths1__mp 27 centrale-maths1__pc 17 centrale-maths1__psi 24 centrale-maths2__mp 29 centrale-maths2__pc 17 centrale-maths2__psi 10
2010
centrale-maths1__mp 19 centrale-maths1__pc 30 centrale-maths1__psi 13 centrale-maths2__mp 32 centrale-maths2__pc 37 centrale-maths2__psi 27
2020 centrale-maths1__psi

40 maths questions

Q1 Independent Events View
Show that $S_n$ and $X_{n+1}$ are independent.
Q2 Probability Generating Functions Explicit computation of a PGF or characteristic function View
Explicitly calculate the generating function $G_{X_1}$ of the random variable $X_1$, where $X_1$ follows a Poisson distribution with parameter $1/2$.
Q3 Probability Generating Functions PGF of sum of independent variables View
Justify that $\forall t \in \mathbb{R}, G_{S_n}(t) = \left(G_{X_1}(t)\right)^n$.
Q4 Poisson distribution View
Show that the random variable $S_n$ follows a Poisson distribution and specify its parameter.
Q5 Poisson distribution View
Verify that, for all $n \in \mathbb{N}^*$, $$n! \left(\frac{2}{n}\right)^n P\left(S_n > n\right) = \mathrm{e}^{-n/2} \sum_{k=1}^{\infty} \frac{n! n^k}{(n+k)!} \left(\frac{1}{2}\right)^k.$$
Q6 Number Theory Divisibility and Divisor Analysis View
Let $n \in \mathbb{N}^*$. Show that for all $k \in \mathbb{N}^*$, $$\left(\frac{n}{n+k}\right)^k \leqslant \frac{n! n^k}{(n+k)!} \leqslant 1.$$
Q7 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
Show that the series of functions $\sum u_k$ where for all $k \in \mathbb{N}^*$, the function $u_k$ is defined on $[0, +\infty[$ by $u_k : x \mapsto (1 + kx)^{-k}(1/2)^k$ is normally convergent on $[0, +\infty[$.
Q8 Sequences and Series Limit Evaluation Involving Sequences View
Deduce that for all $n \in \mathbb{N}^*, \sum_{k \geqslant 1} \left(1 + \frac{k}{n}\right)^{-k} \left(\frac{1}{2}\right)^k$ converges and that $$\lim_{n \rightarrow +\infty} \sum_{k=1}^{\infty} \left(1 + \frac{k}{n}\right)^{-k} \left(\frac{1}{2}\right)^k = 1.$$
Q9 Poisson distribution View
Deduce that, when $n$ tends to $+\infty$, $$P\left(S_n > n\right) \sim \frac{\mathrm{e}^{-n/2}}{n!} \left(\frac{n}{2}\right)^n.$$
Q10 Central limit theorem View
Deduce, using Stirling's formula, that there exists a real $\alpha \in ]0,1[$ such that $P\left(S_n > n\right) = O\left(\alpha^n\right)$.
Q11 Matrices Matrix Norm, Convergence, and Inequality View
Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix. Show that, for all $x \in \mathbb{R}^n$, $$\left\{\begin{array}{l} x \geqslant 0 \Longrightarrow Ax \geqslant 0 \\ x \geqslant 0 \text{ and } x \neq 0 \Longrightarrow Ax > 0. \end{array}\right.$$
Q12 Matrices Matrix Power Computation and Application View
Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix. Show that $\forall k \in \mathbb{N}^*, A^k > 0$.
Q13 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix. Deduce that $\rho(A) > 0$ then show that $\rho\left(\frac{A}{\rho(A)}\right) = 1$.
Q14 Matrices Matrix Power Computation and Application View
Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix. Assume $A$ is diagonalizable over $\mathbb{C}$. Show that, if $\rho(A) < 1$ then $\lim_{k \rightarrow +\infty} A^k = 0$.
Q15 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix satisfying $\rho(A) = 1$. We consider an eigenvalue $\lambda \in \mathbb{C}$ of $A$ with modulus 1 and $x$ an eigenvector associated with $\lambda$. Show that $|x| \leqslant A|x|$.
Q16 Matrices Matrix Norm, Convergence, and Inequality View
Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix satisfying $\rho(A) = 1$. We consider an eigenvalue $\lambda \in \mathbb{C}$ of $A$ with modulus 1 and $x$ an eigenvector associated with $\lambda$. We assume that $|x| < A|x|$. Show that there exists $\varepsilon > 0$ such that $A^2|x| - A|x| > \varepsilon A|x|$.
Q17 Matrices Matrix Power Computation and Application View
Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix satisfying $\rho(A) = 1$. We consider an eigenvalue $\lambda \in \mathbb{C}$ of $A$ with modulus 1 and $x$ an eigenvector associated with $\lambda$. We assume that $|x| < A|x|$ and that there exists $\varepsilon > 0$ such that $A^2|x| - A|x| > \varepsilon A|x|$. We set $B = \frac{1}{1+\varepsilon} A$. Show that for all $k \geqslant 1, B^k A|x| \geqslant A|x|$.
Q18 Matrices Matrix Power Computation and Application View
Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix satisfying $\rho(A) = 1$. We set $B = \frac{1}{1+\varepsilon} A$ for some $\varepsilon > 0$. Determine $\lim_{k \rightarrow +\infty} B^k$.
Q19 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix satisfying $\rho(A) = 1$. We consider an eigenvalue $\lambda \in \mathbb{C}$ of $A$ with modulus 1 and $x$ an eigenvector associated with $\lambda$. Conclude (that 1 is an eigenvalue of $A$).
Q20 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix satisfying $\rho(A) = 1$. Show that $A$ admits a strictly positive eigenvector associated with the eigenvalue 1.
Q21 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix satisfying $\rho(A) = 1$. Show that 1 is the only eigenvalue of $A$ with modulus 1.
One may admit without proof that if $z_1, z_2, \ldots, z_k$ are non-zero complex numbers such that $|z_1 + \cdots + z_k| = |z_1| + \cdots + |z_k|$, then $\forall j \in \llbracket 1, k \rrbracket, \exists \lambda_j \in \mathbb{R}^+$ such that $z_j = \lambda_j z_1$.
Q22 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Throughout part II, $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix satisfying $\rho(A) = 1$. Show that $\dim\left(\ker\left(A - I_n\right)\right) = 1$.
Q23 Matrices Eigenvalue and Characteristic Polynomial Analysis View
By combining the results of sub-parts II.B and II.C, justify that we have proved Proposition 1: If $A \in \mathcal{M}_n(\mathbb{R})$ is a strictly positive matrix, then $\rho(A)$ is a dominant eigenvalue of $A$. The associated eigenspace $\ker\left(A - \rho(A) I_n\right)$ is one-dimensional and is spanned by a strictly positive eigenvector.
Q24 Matrices Matrix Power Computation and Application View
We assume that $A$ is strictly positive and diagonalizable over $\mathbb{C}$. For all $Y \in \mathcal{M}_{n,1}(\mathbb{R})$, for all $p \in \mathbb{N}^*$, we denote $Y_p = \left(\frac{A}{\rho(A)}\right)^p Y$. Let $\lambda \in S = \operatorname{sp}(A) \setminus \{\rho(A)\}$. Let $Y \in \ker\left(A - \lambda I_n\right)$. Show that the sequence $\left(Y_p\right)_{p \in \mathbb{N}^*}$ converges to 0.
Q25 Matrices Matrix Power Computation and Application View
We assume that $A$ is strictly positive and diagonalizable over $\mathbb{C}$. For all $Y \in \mathcal{M}_{n,1}(\mathbb{R})$, for all $p \in \mathbb{N}^*$, we denote $Y_p = \left(\frac{A}{\rho(A)}\right)^p Y$. Let $Y \in \mathcal{M}_{n,1}(\mathbb{R})$ be a positive vector. Show that the sequence $\left(Y_p\right)_{p \in \mathbb{N}^*}$ converges to the projection of $Y$ onto $E_{\rho(A)}(A)$ parallel to $\bigoplus_{\lambda \in S} E_\lambda(A)$. Verify that, if it is non-zero, this latter vector (the projection of $Y$) is strictly positive.
Q26 Matrices Diagonalizability and Similarity View
Justify that for all integer $k \geqslant 1$, $A^k$ is similar in $\mathcal{M}_n(\mathbb{C})$ to a triangular matrix, whose diagonal coefficients we will specify.
Q27 Matrices Matrix Power Computation and Application View
Show that $\lim_{k \rightarrow +\infty} \frac{\operatorname{tr}\left(A^{k+1}\right)}{\operatorname{tr}\left(A^k\right)} = \rho(A)$.
Q28 Discrete Probability Distributions Markov Chain and Transition Matrix Analysis View
Throughout part III, $N$ is a fixed non-zero natural integer and $(X_n)_{n \in \mathbb{N}}$ is a sequence of random variables taking values in $\llbracket 0, N \rrbracket$, forming a homogeneous Markov chain with transition matrix $Q$ where $q_{i,j} = P(X_{n+1} = j \mid X_n = i) > 0$ for all $(i,j) \in \llbracket 0,N \rrbracket^2$. Justify that $\forall i \in \llbracket 0, N \rrbracket, \sum_{j=0}^{N} q_{i,j} = 1$.
Q29 Discrete Probability Distributions Markov Chain and Transition Matrix Analysis View
Throughout part III, $N$ is a fixed non-zero natural integer and $(X_n)_{n \in \mathbb{N}}$ is a sequence of random variables taking values in $\llbracket 0, N \rrbracket$, forming a homogeneous Markov chain with transition matrix $Q$ where $q_{i,j} = P(X_{n+1} = j \mid X_n = i) > 0$ for all $(i,j) \in \llbracket 0,N \rrbracket^2$. We denote $\Pi_n = \begin{pmatrix} P(X_n = 0) \\ \vdots \\ P(X_n = N) \end{pmatrix} \in \mathcal{M}_{N+1,1}(\mathbb{R})$. Justify that, for all $n \in \mathbb{N}^*, \Pi_{n+1} = Q^\top \Pi_n$.
Q30 Discrete Probability Distributions Markov Chain and Transition Matrix Analysis View
Throughout part III, $N$ is a fixed non-zero natural integer and $(X_n)_{n \in \mathbb{N}}$ is a sequence of random variables taking values in $\llbracket 0, N \rrbracket$, forming a homogeneous Markov chain with transition matrix $Q$ where $q_{i,j} = P(X_{n+1} = j \mid X_n = i) > 0$ for all $(i,j) \in \llbracket 0,N \rrbracket^2$. We have $\Pi_{n+1} = Q^\top \Pi_n$. Deduce that the distribution of $X_1$ completely determines the distributions of all random variables $X_n, n \in \mathbb{N}^*$.
Q31 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Throughout part III, $N$ is a fixed non-zero natural integer and $(X_n)_{n \in \mathbb{N}}$ is a homogeneous Markov chain on $\llbracket 0, N \rrbracket$ with transition matrix $Q$ where $q_{i,j} = P(X_{n+1} = j \mid X_n = i) > 0$. We define $a_{i,j}(t) = q_{i,j} \mathrm{e}^{jt}$ and $A(t) = \left(a_{i,j}(t)\right)_{0 \leqslant i,j \leqslant N} \in \mathcal{M}_{N+1}(\mathbb{R})$. Justify that $A(t)$ possesses a dominant eigenvalue $\gamma(t) > 0$.
Q32 Matrices Matrix Power Computation and Application View
Throughout part III, $N$ is a fixed non-zero natural integer and $(X_n)_{n \in \mathbb{N}}$ is a homogeneous Markov chain on $\llbracket 0, N \rrbracket$ with transition matrix $Q$ where $q_{i,j} = P(X_{n+1} = j \mid X_n = i) > 0$. We define $a_{i,j}(t) = q_{i,j} \mathrm{e}^{jt}$, $A(t) = \left(a_{i,j}(t)\right)_{0 \leqslant i,j \leqslant N}$, $z_j(t) = P(X_1 = j)\mathrm{e}^{jt}$, $Z(t) = \begin{pmatrix} z_0(t) \\ \vdots \\ z_N(t) \end{pmatrix}$, and $Y^{(n)}(t) = (A(t))^{n-1} Z(t)$ so that $E\left(\mathrm{e}^{tS_n}\right) = \sum_{j=0}^N Y_j^{(n)}(t)$. Let $\gamma(t)$ be the dominant eigenvalue of $A(t)$. Show that $\lim_{n \rightarrow +\infty} \frac{\ln\left(E\left(\mathrm{e}^{tS_n}\right)\right)}{n} = \lambda(t)$ where $\lambda(t) = \ln(\gamma(t))$.
Q33 Matrices Matrix Power Computation and Application View
Write in Python language a function \texttt{puiss2k} that takes as argument a square matrix $M$ and a natural integer $k$ and returns the matrix $M^{2^k}$ by performing $k$ matrix products. One may exploit the fact that $M^{2^{k+1}} = M^{2^k} M^{2^k}$.
Q34 Matrices Matrix Power Computation and Application View
Explain what the Python function \texttt{maxSp} defined by: \begin{verbatim} def maxSp(Q:np.ndarray, k:int, t:float) -> float: n = Q.shape[1] E = np.exp(t * np.array(range(n))) A = Q * E B = puiss2k(A, k) C = np.dot(A, B) return C.trace() / B.trace() \end{verbatim} does.
Q35 Continuous Probability Distributions and Random Variables Probability Inequality and Tail Bound Proof View
We define, for all $x \in \mathbb{R}$, $\lambda^*(x) = \sup_{t \geqslant 0}(tx - \lambda(t))$. We admit that the convergence of the sequence of functions $\left(t \mapsto \frac{\ln\left(E\left(\mathrm{e}^{tS_n}\right)\right)}{n}\right)_{n \in \mathbb{N}^*}$ towards $t \mapsto \ln(\gamma(t))$ is uniform on $\mathbb{R}^+$. Let $\varepsilon > 0$. Show that there exists a rank $n_0 \in \mathbb{N}^*$ such that, for all $t \in \mathbb{R}^+$ and for all $n \in \mathbb{N}^*$, $$n \geqslant n_0 \Longrightarrow \ln\left(E\left(\mathrm{e}^{tS_n}\right)\right) \leqslant n(\lambda(t) + \varepsilon).$$
Q36 Continuous Probability Distributions and Random Variables Probability Inequality and Tail Bound Proof View
We define, for all $x \in \mathbb{R}$, $\lambda^*(x) = \sup_{t \geqslant 0}(tx - \lambda(t))$. Let $\varepsilon > 0$ and $n_0$ as in Q35. Using Markov's inequality applied to the random variable $e^{tS_n}$, show that for $a > 1$, $n \geqslant n_0$ and $t \geqslant 0$, $$P\left(S_n \geqslant nam\right) \leqslant \mathrm{e}^{-ntam} \mathrm{e}^{n(\lambda(t) + \varepsilon)}.$$
Q37 Continuous Probability Distributions and Random Variables Probability Inequality and Tail Bound Proof View
We define, for all $x \in \mathbb{R}$, $\lambda^*(x) = \sup_{t \geqslant 0}(tx - \lambda(t))$. Let $\varepsilon > 0$ and $n_0$ as in Q35. Deduce that for $n \geqslant n_0$, $$P\left(S_n \geqslant nam\right) \leqslant \mathrm{e}^{-n\left(\lambda^*(am) - \varepsilon\right)}.$$
Q38 Continuous Probability Distributions and Random Variables Probability Inequality and Tail Bound Proof View
Give a concrete meaning to $m = \lim_{n \rightarrow +\infty} \frac{1}{n} E(S_n)$ in relation to the industrial process studied and interpret the inequality $P\left(S_n \geqslant nam\right) \leqslant \mathrm{e}^{-n(\lambda^*(am) - \varepsilon)}$. One may establish an intuitive link with the law of large numbers.
Q39 Proof Deduction or Consequence from Prior Results View
We have two finite sequences of real numbers $0 = t_1 < t_2 < \cdots < t_K$ ($K \geqslant 2$) and $x_1 < x_2 < \cdots < x_L$ ($L \geqslant 2$). The formula from question 32 applied at $t_i$ with $n$ sufficiently large allows us to estimate $\lambda(t_i)$ by an approximate value $\hat{\lambda}(t_i)$. Justify that for all $i \in \{1, \ldots, L\}$, $$\hat{\lambda}^*\left(x_i\right) = \max_{1 \leqslant j \leqslant K} \left(t_j x_i - \hat{\lambda}\left(t_j\right)\right)$$ constitutes a reasonable approximate value of $\lambda^*\left(x_i\right)$.
Q40 Proof Computation of a Limit, Value, or Explicit Formula View
Using Table 1 below, give an approximate bound for the value of $m$ and the value of a real number $h > 0$ such that there exists a rank $n_0 \in \mathbb{N}^*$ satisfying for all $n \geqslant n_0$, $$P\left(S_n > 1{,}1 \times nm\right) \leqslant \mathrm{e}^{-nh}.$$
$x_i$4,504,554,604,654,70
$\hat{\lambda}^*(x_i)$$4{,}1 \times 10^{-12}$$4{,}1 \times 10^{-12}$$4{,}1 \times 10^{-12}$$4{,}1 \times 10^{-12}$$4{,}1 \times 10^{-12}$
$x_i$4,754,804,854,904,95
$\hat{\lambda}^*(x_i)$$5{,}1 \times 10^{-4}$$5{,}5 \times 10^{-3}$$1{,}1 \times 10^{-2}$$1{,}6 \times 10^{-2}$$2{,}1 \times 10^{-2}$
$x_i$5,005,055,105,155,20
$\hat{\lambda}^*(x_i)$$2{,}6 \times 10^{-2}$$3{,}1 \times 10^{-2}$$3{,}6 \times 10^{-2}$$4{,}1 \times 10^{-2}$$4{,}6 \times 10^{-2}$
$x_i$5,255,305,355,405,45
$\hat{\lambda}^*(x_i)$$5{,}1 \times 10^{-2}$$5{,}6 \times 10^{-2}$$6{,}1 \times 10^{-2}$$6{,}6 \times 10^{-2}$$7{,}1 \times 10^{-2}$