Source Latex: Cours de mathématiques en Terminale S


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Type: Cours
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Description
Annexe du cours de mathématiques et probabilités: théorème de Moivre-Laplace, approximation de la loi binomiale
Niveau
Terminale S
Table des matières
  • Table de la fonction intégrale (ou de répartition) de la loi normale cenrée réduite
  • Théorème de Moivre-Laplace: approximation de la loi binomiale
Mots clé
Cours de mathématiques, Moivre-Laplace, approximation loi binomile, TS, terminale S
Voir aussi:

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Source Latex du cours de mathématiques

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    pdfauthor={Yoann Morel},
    pdfsubject={Cours de math�matiques: Probabilit�s continues},
    pdftitle={Probabilit�s - Lois continues },
    pdfkeywords={Math�matiques, TS, terminale, S, 
      probabilit�, probabilit�s, loi binomiale, 
      loi � densit�, loi continue, loi normale, 
      loi uniforme
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\vspace*{-0.5cm}


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\ct{\bf Extrait de la table de la fonction int�grale de la loi
  normale centr�e r�duit $\mathcal{N}(0;1)$ }

\vspd
La loi normale centr�e r�duite $\mathcal{N}(0;1)$ est la loi de
probabilit� de densit� 
$\dsp f(x)=\dfrac{1}{\sqrt{2\pi}}\,e^{-\frac{x^2}{2}}$

\vspd
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\[
\Pi(t)=P\lp X\leqslant t\rp=\int_{-\infty}^t f(x)\,dx
\]
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\begin{tabular}{|>{\columncolor{lightgray}}c|*{10}{c|}}\hline
\rowcolor{lightgray}                                         
$t$ & $0,0 0 $ & $0,0 1 $ & $0,0 2 $ & $0,0 3 $ & $0,0 4 $ & $0,0 5 $ & $0,0 6 $ & $0,0 7 $ & $0,0 8 $ & $0,0 9 $ \\ \hline                                                                                                     
0.0 & 0.5 & 0.5039 & 0.5079 & 0.5119 & 0.5159 & 0.5199 & 0.5239 & 0.5279 & 0.5318 & 0.5358 \\ \hline            
0.1 & 0.5398 & 0.5437 & 0.5477 & 0.5517 & 0.5556 & 0.5596 & 0.5635 & 0.5674 & 0.5714 & 0.5753 \\ \hline         
0.2 & 0.5792 & 0.5831 & 0.587 & 0.5909 & 0.5948 & 0.5987 & 0.6025 & 0.6064 & 0.6102 & 0.614 \\ \hline           
0.3 & 0.6179 & 0.6217 & 0.6255 & 0.6293 & 0.633 & 0.6368 & 0.6405 & 0.6443 & 0.648 & 0.6517 \\ \hline           
0.4 & 0.6554 & 0.659 & 0.6627 & 0.6664 & 0.67 & 0.6736 & 0.6772 & 0.6808 & 0.6843 & 0.6879 \\ \hline            
0.5 & 0.6914 & 0.6949 & 0.6984 & 0.7019 & 0.7054 & 0.7088 & 0.7122 & 0.7156 & 0.719 & 0.7224 \\ \hline          
0.6 & 0.7257 & 0.729 & 0.7323 & 0.7356 & 0.7389 & 0.7421 & 0.7453 & 0.7485 & 0.7517 & 0.7549 \\ \hline          
0.7 & 0.758 & 0.7611 & 0.7642 & 0.7673 & 0.7703 & 0.7733 & 0.7763 & 0.7793 & 0.7823 & 0.7852 \\ \hline          
0.8 & 0.7881 & 0.791 & 0.7938 & 0.7967 & 0.7995 & 0.8023 & 0.8051 & 0.8078 & 0.8105 & 0.8132 \\ \hline          
0.9 & 0.8159 & 0.8185 & 0.8212 & 0.8238 & 0.8263 & 0.8289 & 0.8314 & 0.8339 & 0.8364 & 0.8389 \\ \hline         
1.0 & 0.8413 & 0.8437 & 0.8461 & 0.8484 & 0.8508 & 0.8531 & 0.8554 & 0.8576 & 0.8599 & 0.8621 \\ \hline         
1.1 & 0.8643 & 0.8665 & 0.8686 & 0.8707 & 0.8728 & 0.8749 & 0.8769 & 0.8789 & 0.8809 & 0.8829 \\ \hline         
1.2 & 0.8849 & 0.8868 & 0.8887 & 0.8906 & 0.8925 & 0.8943 & 0.8961 & 0.8979 & 0.8997 & 0.9014 \\ \hline         
1.3 & 0.9031 & 0.9049 & 0.9065 & 0.9082 & 0.9098 & 0.9114 & 0.913 & 0.9146 & 0.9162 & 0.9177 \\ \hline          
1.4 & 0.9192 & 0.9207 & 0.9221 & 0.9236 & 0.925 & 0.9264 & 0.9278 & 0.9292 & 0.9305 & 0.9318 \\ \hline          
1.5 & 0.9331 & 0.9344 & 0.9357 & 0.9369 & 0.9382 & 0.9394 & 0.9406 & 0.9417 & 0.9429 & 0.944 \\ \hline          
1.6 & 0.9452 & 0.9463 & 0.9473 & 0.9484 & 0.9494 & 0.9505 & 0.9515 & 0.9525 & 0.9535 & 0.9544 \\ \hline         
1.7 & 0.9554 & 0.9563 & 0.9572 & 0.9581 & 0.959 & 0.9599 & 0.9607 & 0.9616 & 0.9624 & 0.9632 \\ \hline          
1.8 & 0.964 & 0.9648 & 0.9656 & 0.9663 & 0.9671 & 0.9678 & 0.9685 & 0.9692 & 0.9699 & 0.9706 \\ \hline          
1.9 & 0.9712 & 0.9719 & 0.9725 & 0.9731 & 0.9738 & 0.9744 & 0.975 & 0.9755 & 0.9761 & 0.9767 \\ \hline          
2.0 & 0.9772 & 0.9777 & 0.9783 & 0.9788 & 0.9793 & 0.9798 & 0.9803 & 0.9807 & 0.9812 & 0.9816 \\ \hline         
2.1 & 0.9821 & 0.9825 & 0.9829 & 0.9834 & 0.9838 & 0.9842 & 0.9846 & 0.9849 & 0.9853 & 0.9857 \\ \hline         
2.2 & 0.986 & 0.9864 & 0.9867 & 0.9871 & 0.9874 & 0.9877 & 0.988 & 0.9883 & 0.9886 & 0.9889 \\ \hline           
2.3 & 0.9892 & 0.9895 & 0.9898 & 0.99 & 0.9903 & 0.9906 & 0.9908 & 0.9911 & 0.9913 & 0.9915 \\ \hline           
2.4 & 0.9918 & 0.992 & 0.9922 & 0.9924 & 0.9926 & 0.9928 & 0.993 & 0.9932 & 0.9934 & 0.9936 \\ \hline           
2.5 & 0.9937 & 0.9939 & 0.9941 & 0.9942 & 0.9944 & 0.9946 & 0.9947 & 0.9949 & 0.995 & 0.9952 \\ \hline          
2.6 & 0.9953 & 0.9954 & 0.9956 & 0.9957 & 0.9958 & 0.9959 & 0.996 & 0.9962 & 0.9963 & 0.9964 \\ \hline          
2.7 & 0.9965 & 0.9966 & 0.9967 & 0.9968 & 0.9969 & 0.997 & 0.9971 & 0.9971 & 0.9972 & 0.9973 \\ \hline          
2.8 & 0.9974 & 0.9975 & 0.9975 & 0.9976 & 0.9977 & 0.9978 & 0.9978 & 0.9979 & 0.998 & 0.998 \\ \hline           
2.9 & 0.9981 & 0.9981 & 0.9982 & 0.9983 & 0.9983 & 0.9984 & 0.9984 & 0.9985 & 0.9985 & 0.9986 \\ \hline         
\end{tabular}                                                                                                   

\vspt\noindent{\bf Table pour les grandes valeurs de $t$}

\hspace*{-1.2cm}
\begin{tabular}{|c|*{10}{c|}}\hline
$t$ & $3.0$ & $3.1$ & $3.2$ & $3.3$ & $3.4$ & $3.5$ & $3.6$ & $3.8$ & $4.0$ & $4.5$ \\ \hline
$\Pi(t)$ &  $ 0.99865 $ & $ 0.99903 $ & $ 0.99931 $ & $ 0.99951 $ & $ 0.99966 $ & $ 0.99976 $ & $ 0.99984 $ & $ 0.99992 $ & $ 0.99996 $ & $ 0.99999 $ \\ \hline
\end{tabular}


%\section*{{\LARGE Annexe:} Approximation d'une loi binomiale par une
%  loi Normale - Th�or�me de Moivre-Laplace} 

\noindent
{\bf{\LARGE\ul{Annexe:}}\hspace{0.2cm}
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{\Large Approximation d'une loi binomiale par une
  loi Normale - Th�or�me de Moivre-Laplace} 
\enmp}

\vspace{0.4cm}

Le th�or�me de Moivre-Laplace affirme que, 
pour $n$ suffisamment grand, on peut remplacer les probabilit�s
associ�es � la loi binomiale $\mathcal{B}(n;p)$ par celles de la loi
normale $\mathcal{N}(\mu;\sigma^2)$ avec $\mu=np$ et $\sigma=\sqrt{npq}$  


\vspd
En pratique, on approche les probabilit�s de la loi binomiale par
celles de la loi normale lorsque 
\ct{\fbox{$n\geqslant 50$,\quad $np\geqslant 5$\quad et\quad $nq\geqslant 5$}}. 

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\end{document}

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