stats_17_10_24.py

Created by famille-bvc

Created on October 17, 2024

2.45 KB


** Notes **
_
X = Moyenne

-----

** Ecart type : 
       X^2   _      ni*xi^2  _
S^2 = ----- - X^2 ou -------- -X^2
       Nbx             Nbx

σ = S(X) = sqrt(S^2) 

Variance V=S^2

-----

** covariance :
            __   _   _
  cov(x,y)= xy - x * y 
  
** coefficient de correlation :
  
  p = cov(x,y) / σx*σy

si proche de 1 -> ajust affine

-----

** Regression lineaire :
  
  p = ax + b
  
         cov(x,p)
avec a = --------
          S^2(x)
    _       _
b = p - a * x

** Regression exponentielle :
  
  y = a^x * b

ex : p = ln(y)=0.4x+1.8
-> y = e(a)^x * e(b)

*********************************
** Loi normale centree reduite **

à la base -> N(0, 1)

*********************************
**      Loi exponentielle      **

P(T <= t) = 1 - e(-)

demie vie :
                    ln(2)
e(-λt0)=1/2 => t0 = -----
                      λ
                      
esperance :
  
E(T) = 1/λ

*********************************
**        Loi binomiale        **

proba que n fois un evenement 
atteigne la proba voulue
-> on cheche n

P: la proba que l on veut
qn: la proba que ça arrive

n = ln(1-P)/ln(1-qn)

-----

si proba binomiale avec :
  
  n > 30    np >= 5    nq >= 5
  
-> approx par loi normale
N(np; sqrt(npq))
_
X = np et S = sqrt(npq)

*********************************
**        Loi de Student       **

A la base -> P(T>=t)=x

donc P(T<=t) = x
  => P(T>=t) = 1-x
  
donc P(-t<=T<=t) = x
  => P(T>=t) = (1-x)/2
  
*********************************
**         Loi de Khi2         **

A la base -> P(X>=x)=y

donc P(X<=x) = y
  => P(X>=x) = 1-y
  
Quand v > 100:
une loi de khi2 peut etre 
approchee par loi N(v;sqrt(2v))

*********************************
**   Intervalle de confiance   **
  
*Si n>30 -> loi normale ecart r:
  
Tout d abord calculer S^2c :
  
S^2c = (n)/(n-1) * S^2
Sc = sqrt(S^2c)
--
  
a% = 1 - seuil de confiance %
za/2 = voir tableau en f de a

        _          Sc    _
ICa% = [X- za/2 *------; X + ...]
                 sqrt(n)
                  
*Sinon n < 30 -> loi de student:

Tout d abord calculer Sc :
  
Sc = sqrt((n)/(n-1))* S
--

a%= (1 - seuil de confiance)/2 %
ta;n-1 = voir tableau en f de a
  
      _            Sc    _
ICa%=[X- ta;n-1 *------; X + ...]
                 sqrt(n)
                  
-----

calcul taille de l echantillon
pour un intervalle donne :
  
        2 * za/2 * SC
ICa% = ---------------
           sqrt(n)
           
resoudre l equation :
  
ICa% = intervalle donne 
        pour avoir n
        

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