Finite Population Sampling without Replacement

Personal note of finite population sampling

First moment Population: $ \Omega = \{ x_1, \dots, x_N \} $ Collection of $n$-samples: $\mathcal{S} = \{ s \in \Omega^n \mid \forall i,j \in s, i \ne j \} $ Collection of $n$-samples containing $x$: $ \mathcal{S}_x = \{ s \in \mathcal{S} \mid x \in s \} $ Observe that $ |\mathcal{S}_x| = \binom{N-1}{n-1} $. Let population mean be zero. $\mu = 0$, i.e. $ \sum_{i = 1}^N x_i = 0 $ [Read More]

Borel Cantelli Exercise 2019

I intend to post this for a Borel-Cantelli lemma exercise on Math.SE.

The target event is ${\exists i_0 \in \Bbb{N} : \forall i \ge i_0, X_i = 1}$, whose complement is

$$ {\forall i_0 \in \Bbb{N} : \exists i \ge i_0, X_i > = 0} = \limsup_i {X_i = 0}. $$

To apply Borel-Cantelli, one has to determine whether $\sum_i P(X_i = 0)<+\infty$.

Weak LLN Practice

My intended answer to a weak LLN problem on Math.SE. Problem: Suppose $(X_n)$ is a sequence of r.v’s satisfying $P(X_n=\pm\ln (n))=\frac{1}{2}$ for each $n=1,2\dots$. I am trying to show that $(X_n)$ satisfies the weak LLN. The idea is to show that $P(\overline{X_n}>\varepsilon)$ tends to 0, but I am unsure how to do so. My solution: As in the accepted answer in OP’s previous question https://math.stackexchange.com/q/3021650/290189, I’ll assume the independence of $(X_n)$. [Read More]

Mesurabilité des réalisations trajectorielles

$X: \omega \mapsto X(\cdot, \omega) \in \mathcal{M}((\Omega, \mathcal{A}), (\CO(\Bbb{T},\R), \Bor{\CO}))$

Notations Supposons toutes les notations dans Espace de trajectoires. Problématique La mesurabilité de l’application dans le sous-titre est basée sur l’égalité suivante. $$ \Bor{\R}{\OXT} \cap \CO = \Bor{\CO} $$ J’ai passé quatres heures pour comprendre pourquoi ça entraîne la mesurabilité ? pourquoi l’égalité elle-même est vraie ? Réponses Mesurabilité de la trace sur $\CO$ de $\Bor{\R}{\OXT}$ A la première lecture, je ne connaisais même pas la définition de la trace d’une tribu sur un emsemble. [Read More]

2018-10-04 Seminar Notes

I jotted down only a few keywords that might be reusable. I didn’t understand any of the talks. Functional Data Analysis Goal: predict equipment temperature Tools: Fourier coefficients (trigo ones), followed by discretisation, min-error estimation, cross-validation 10-folds, $R^2$ adjusted ?, MAE, MSPE Comparison with non-functional data Tolérancement Thème : Traiter les incertitudes sur les dimensions des pièces de l’avion Objectif : établir une modélisation mathématiques construire un virtual twin de l’avion Outils : Modèle de variabilité Modèle d’assemblage $\text{airbus}: Y = \sum_{i = 1}^n a_iX_i? [Read More]

Measures Are Regular

Some remarks on constructing the $\sigma$-algebra

Problem To show that a measure $\mu$ defined on a metric space $(S,d)$ is regular. outer regularity: approximation by inner closed sets inner regularity: approximation by outer open sets Discussion Since this problem involves all borel sets $A \in \mathcal{B}(S)$, the direct way $\forall A \in \mathcal{B}(S), \dots$ won’t work. We have to use the indirect way: denote $$\mathcal{C} = \lbrace A \in \mathcal{B}(S) \mid \mathinner{\text{desired properties}} \dots \rbrace.$$ Show that [Read More]

Espace de trajectoires

Comparaison des références

Tribu produit source symbole engendrée par Prof $\Er{\OXT}$ $\mathcal{C}_0 = \Big\lbrace \lbrace f \in E^\Bbb{T} \mid f(t) \in B \rbrace \bigm\vert t \in \Bbb{T}, B \in \Er \Big\rbrace$ $\mathcal{C}_1 = \Big\lbrace \lbrace f \in E^\Bbb{T} \mid f(t_i) \in B_i \forall i \in \lbrace 1,\dots,n \rbrace \rbrace \newline \bigm\vert t_j \in \Bbb{T}, B_j \in \Er \forall j \in \lbrace 1,\dots,n \rbrace, n \in \N^* \Big\rbrace$ Meyre $\bigotimes_{t \in \Bbb{T}} \Er$ des cylindres $C = \prod_{t \in \Bbb{T}} A_t$ d’ensembles mesurables $A_t \in \Er$ de dimension finie $\card{\lbrace t \in \Bbb{T} \mid A_t \neq E \rbrace} < \infty$ Je trouve $\Er{\OXT}$ plus court à écrire, tandis que $\bigotimes_{t \in \Bbb{T}} \Er$ est plus flexible. [Read More]

$\pi$–$\lambda$ Theorem

Monotone Class Lemma

Statement Slogan version $$\sigma = \pi + \lambda$$ $\sigma$ $\pi$ $\lambda$ “sum” “product” “limit” universe nonempty universe complement complement countable union finite intersection disjoint countable union A $\sigma$-algebra is a $\pi$-system and a $\lambda$-system, and vice versa. Wiki version A $\lambda$-system is a synonym of a Dykin system. $$\mathcal{P} \subseteq \mathcal{D} \Longrightarrow \sigma(\mathcal{P}) \subseteq \mathcal{D}$$ Given a $\pi$-system contained in a $\lambda$-system. Then the $\sigma$-algebra generated by the $\pi$-system is also contained in the $\lambda$-system. [Read More]