x<- c(1,2,3,4,5,6)
print(mean(x))[1] 3.5Dictionnaire Rmarkdown
Écrire du code R
On peut insérer une cellule de code R à l’aide de Ctrl + Alt + I (Cmd + Option + I pour macOS).
Écrire des mathématiques avec Rmarkdown
On peut écrire des symboles mathématiques à l’aide du symbole $. Une expression mathématique doit commencer et terminer par $. Par exemple $\alpha = 0.05\%$ devient \(\alpha = 0.05\%\). Dans la suite de ce dictionnaire se trouve des commandes usuelles pour répondre aux TD et TP.
Mathématiques générales
\(x\)
$x$\(X\)
$X$\(A_b\)
$A_b$\(A^b\)
$A^b$\(A+B\)
$A+B$\(A-B\)
$A-B$\(A\times B\)
$A\times B$\(\frac{A}{B}\)
$\frac{A}{B}$\(X_i\)
$X_i$\((x_1,\dots,x_n)\)
$(x_1,\dots,x_n)$\((X_1,\dots,X_n)\)
$(X_1,\dots,X_n)$Théorie des tests
\(\alpha\)
$\alpha$\(\beta\)
$\beta$\(\mu\)
$\mu$\(\mu = \mathbb{E}[X]\)
$\mu = \mathbb{E}[X]$\(\sigma\)
$\sigma$\(\sigma^2\)
$\sigma^2$\(\sigma^2 = \mathbb{V}[X]\)
$\sigma^2 = \mathbb{V}[X]$\(X \sim \mathcal{N}(\mu,\sigma^2)\)
$X \sim \mathcal{N}(\mu,\sigma^2)$\(\bar{X}_n\)
$\bar{X}_n$\(\sum_{i=1}^n X_i\)
$\sum_{i=1}^n X_i$\(\bar{X}_n = \frac{1}{n}\sum_{i=1}^n X_i\)
$\bar{X}_n = \frac{1}{n}\sum_{i=1}^n X_i$\(S^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i-\bar{X}_n)^2\)
$S^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i-\bar{X}_n)^2$\(H_0\)
$H_0$\(H_1\)
$H_1$\(\mu=\mu_0\)
$\mu=\mu_0$\(\mu>\mu_0\)
$\mu>\mu_0$\(\mu\geq\mu_0\)
$\mu\geq\mu_0$\(\mu<\mu_0\)
$\mu<\mu_0$\(\mu\leq\mu_0\)
$\mu\leq\mu_0$\(\mu\neq\mu_0\)
$\mu\neq\mu_0$\(\alpha = 0.05\%\)
$\alpha = 0.05\%$\(\mathbb{P}(A)\)
$\mathbb{P}(A)$\(\mathbb{P}(\mu \in IC)\)
$\mathbb{P}(\mu \in IC)$\(\sqrt{n}\)
$\sqrt{n}$\(\sqrt{n}\frac{(\bar{X}_n-\mu)}{S} \sim \mathcal{N}(0,1)\)
$\sqrt{n}\frac{(\bar{X}_n-\mu)}{S} \sim \mathcal{N}(0,1)$\(\sqrt{n}\frac{(\bar{X}_n-\mu)}{S} \sim T(n-1)\)
$\sqrt{n}\frac{(\bar{X}_n-\mu)}{S} \sim T(n-1)$\(\mathbb{P}(\sqrt{n}\frac{(\bar{X}_n-\mu)}{S} \leq q_{1-\alpha} )= 1-\alpha\)
$\mathbb{P}(\sqrt{n}\frac{(\bar{X}_n-\mu)}{S} \leq q_{1-\alpha} )= 1-\alpha$\(\mathbb{P}(q_{1-\alpha/2} \leq \sqrt{n}\frac{(\bar{X}_n-\mu)}{S} \leq q_{1-\alpha/2} )= 1-\alpha\)
$\mathbb{P}(q_{1-\frac{\alpha}{2}} \leq \sqrt{n}\frac{(\bar{X}_n-\mu)}{S} \leq q_{1-\frac{\alpha}{2}} )= 1-\alpha$\(IC_{\alpha}(\mu) = [\bar{X}_n-\frac{S}{\sqrt{n}} q_{1-\frac{\alpha}{2}} ,\bar{X}_n+\frac{S}{\sqrt{n}} q_{1-\frac{\alpha}{2}}]\)
$IC_{\alpha}(\mu) = [\bar{X}_n-\frac{S}{\sqrt{n}} q_{1-\frac{\alpha}{2} ,\bar{X}_n+\frac{S}{\sqrt{n}} q_{1-\frac{\alpha}{2}]$\([\bar{x}_n-\frac{s}{\sqrt{n}} q_{1-\frac{\alpha}{2}} ,\bar{x}_n+\frac{s}{\sqrt{n}} q_{1-\frac{\alpha}{2}}]=[73.37;75.47]\)
$[\bar{x}_n-\frac{s}{\sqrt{n}} q_{1-\frac{\alpha}{2}} ,\bar{x}_n+\frac{s}{\sqrt{n}} q_{1-\frac{\alpha}{2}}]=[73.37;75.47]$\(p_c=\mathbb{P}_{H_0}(|T_n|>|t_n|)\)
$p_c=\mathbb{P}_{H_0}(|T_n|>|t_n|)$Modèles linéaires
\(E_i \overset{i.i.d.}\sim \mathcal{N}(0,\sigma^2)\)
$E_i \overset{i.i.d.}\sim \mathcal{N}(0,\sigma^2)$\(Y_i = \beta + \sum_{j=1}^p\alpha_j x_{i,j} + E_i, \quad E_i\overset{i.i.d.}\sim{\cal N}(0,\sigma^2)\)
$Y_i = \beta + \sum_{j=1}^p\alpha_j x_{i,j} + E_i, \quad E_i\overset{i.i.d.}\sim{\cal N}(0,\sigma^2)$\(Y=X\theta+E\)
$Y=X\theta+E$\(Y =\begin{bmatrix} Y_1\\ Y_2\\ \vdots \\ Y_n\end{bmatrix}\)
$Y =\begin{bmatrix}
Y_1 \\
Y_2 \\
\vdots\\
Y_n
\end{bmatrix}$\(X = \begin{bmatrix}1 & x_{1,1} & \ldots & x_{p, 1}\\1 & x_{1, 2} & \ldots & x_{p, 2}\\\vdots & \vdots & & \vdots \\1 & x_{1, n} & \ldots & x_{p, n}\end{bmatrix}\)
$X = \begin{bmatrix}
1 & x_{1,1} & \ldots & x_{p, 1}\\
1 & x_{1, 2} & \ldots & x_{p, 2}\\
\vdots & \vdots & & \vdots \\
1 & x_{1, n} & \ldots & x_{p, n}
\end{bmatrix}$\(\theta = \begin{bmatrix} \beta\\ \alpha_1\\\alpha_2\\ \vdots \\ \alpha_{p}\end{bmatrix}\)
$\theta = \begin{bmatrix}
\beta \\
\alpha_1\\
\alpha_2\\
\vdots \\
\alpha_{p}
\end{bmatrix}$\(E = \begin{bmatrix}E_1\\E_2\\ \vdots \\ E_n\end{bmatrix}\)
$E = \begin{bmatrix}
E_1 \\
E_2 \\
\vdots \\
E_n
\end{bmatrix}$\(E \sim \mathcal{N}\left(\vec{0}_n,\,\sigma^2 I_n\right)\)
$E \sim \mathcal{N}\left(\vec{0}_n,\,\sigma^2 I_n\right)$