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Mathematical prerequisities - Coggle Diagram
Mathematical prerequisities
Linear Vector space
Unitary (if accepts scalar product)
Norm = physical lengths
Hilbert space
Complete
Every Cauchy sequence here converges to a vector
Least action principle (Hamilton)
Probabilities
La probabilité de l'aléatoire
La probabilité-incertitude
tribu F
$$L'ensemble\ vide\ \emptyset\ est\ dans\ \mathcal{F}$$
$$ if\ we\ have\ a\ finite\ suite\ ,\ A_{1},A_{2},...,A_{n}\ then\ their\ reunion\ \bigcup_{n\geq 1}A_{n} \ is\ in\ \mathcal{F}$$
$$if\ A \in F \ then \ A^c \in \mathcal{F} $$
Definition
$$The\ certain\ event\ \mathcal{P}(\mathcal{A})\ =\ 1$$
Denombrable addivity axiom
$$Having\ a\ tribu\ \mathcal{F},\ having\ many\ subsets\ of\ \omega,\ a\ probability\ \mathcal{P}\ \in [0,1],\ then\ $$$$a\ process\ assocying\ every\ event\ \mathcal{A}\ a\ number\ \mathcal{P}(\mathcal{A})\ between\ 0\ and\ 1\ $$$$called\ probability\ of\ \mathcal{A},and\ satisfies\ the\ axioms :$$
Continuous Proba
Universe is no linger denumbrable
Borelian tribu for real numbers
Variables aléatoires
Repartition Function
$$F_x(x)\ =\ P_x(]-\infty ,x])\ =\ P(X\leq x)$$
$$Exponential\ :\ F(x)\ =\ 1-e^{-\lambda x}$$
has its own
density
!!
Processus aléatoire
Bernoulli
Poisson
Markov
Loi de proba
$$uniform\ \mathcal{U}(a,b)\ :\ \frac{1}{b-a}$$
$$Exponential\ \mathcal{E}(\lambda):\ \mathcal{f}(x)\ =\ \lambda e^{-\lambda x}$$
$$Gamma\ G(\alpha,\lambda)\ :\ \frac{\lambda ^{\alpha}}{\Gamma(\alpha)} x^{\alpha -1} e^{-\lambda x}$$
$$Normale\ \mathcal{N}(m,\sigma^2)\ :\ \frac{1}{\sigma \sqrt{2 \pi}} e^{- \frac{(x-m)^2}{2 \sigma^2}}$$
$$Weibull \ f(x)\ =\ \lambda \alpha x^{\alpha - 1} e^{- \lambda x^{\alpha - 1}}$$
$$\alpha\ \lt\ 1\ La\ pièce \ s'améliore\ au\ fil\ du\ temps$$
$$\alpha\ =\ 1\ La\ pièce \ ne\ subit\ pas\ de\ modifications $$
$$\alpha\ \gt\ 1\ La\ pièce \ en\ vieillissement\ accéléré$$
Intermediate axis theorem