### vect alea

parent ea02d6a8
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 ... ... @@ -48,7 +48,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ "# Densité, fonction de réparition, quantiles" "## Densité, fonction de réparition, quantiles" ] }, { ... ... @@ -199,7 +199,7 @@ "metadata": {}, "source": [ "***Exo:*** laquelle de ces deux représentations correpond à la fonction de répartition, à savoir: \n", "\t$x \\to P[ X\\leq x]$\n" "\t$x \\to P[ X\\leq x]$\n" ] }, { ... ... @@ -220,7 +220,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ "# Conseil sur les arguments facultatifs\n", "## Conseil python: les arguments facultatifs\n", "\n", "Considérons un appel d'une fonction de scipy :" ] ... ... @@ -267,7 +267,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ "# lois continues classiques" "## lois continues classiques" ] }, { ... ... @@ -284,7 +284,7 @@ "\n", "\n", "\n", "***Proposition :*** si $x\\to f(x)$ est la densité d'une va $X$, alors la densité de $\\sigma X + \\mu$ est:\n", "***Proposition :*** si $x\\to f(x)$ est la densité d'une va $X$, alors la densité de $\\sigma X + \\mu$ est:\n", "$$\n", " \\frac 1 \\sigma \\ f \\ \\Big( \\frac{ x-mu} \\sigma \\Big)\n", "$$\n", ... ... @@ -292,21 +292,18 @@ "\n", "*Astuce:* Pour ne pas s'encombrer la mémoire, retenez uniquement les densités des lois dans le cas $\\mu=0$ et $\\sigma=1$. Par exemple: \n", "\n", "$$\n", "\\begin{array}{cc}\n", "a & b \\\\\n", "\\end{array}\n", "$$\n", "\n", "\n", "| nom de la loi | simplifié | complète|\n", "| nom de la loi |        simplifié         |        complète         |\n", "|---------------|-----------| --------|\n", "| exponentielle | $\\exp(x) 1_{\\{x>0\\}}$ | $\\frac {1} {\\sigma} \\exp(- \\frac {x-\\mu } {\\sigma} )$ |\n", "\n", "| exponentielle | $e^{-x} 1_{\\{x>0\\}}$ | $\\frac {1} {\\sigma} \\exp(- \\frac {x-\\mu } {\\sigma} )$ |\n", "| normale | $\\frac {1} { \\sqrt {2 \\pi}} e^{-\\frac{1}{2} x^2}$ | ... |\n", "\n", "\n", " | normale | $\\frac {1} { \\sqrt {2 \\pi}} exp -x^2$ | ... |\n", "\n", "\n", "| aaa | bbb |\n", "|---|---|\n", "| $a$ | $b$ |\n", "\n", "\n", "\n", "\n", ... ... @@ -316,7 +313,7 @@ "$$\n", " \\mathbf E[\\phi( \\sigma X + mu )] = \\int \\phi( \\sigma x + \\mu) \\ f(x) \\ dx \n", "$$\n", "On effectue le changement de variable $\\sigma x + \\mu \\to y$ \n", "On effectue le changement de variable $\\sigma x + \\mu \\to y$ \n", "$$\n", " \\mathbf E[\\phi( \\sigma X + mu )] = \\int \\phi( y) \\ f(... ) \\ dy ... \n", "$$\n", ... ... @@ -471,7 +468,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ "# Lois à Queues lourdes\n", "## Lois à Queues lourdes\n", "\n", "Une loi est dites à queue lourde lorsque les v.a qui ont cette loi peuvent prendre, de temps en temps, des grandes valeurs positives ou négatives. Elle servent à modéliser des évènements rare et violent (ex: crue d'un fleuve). " ] ... ... @@ -598,7 +595,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ "# Relations entre les lois\n", "## Relations entre les lois\n", "\n", "\n", "Il est important de retenir les relations entre les principales lois, l'image ci-dessous peut vous aider à cela. Sinon, sur cette [page](http://www.math.wm.edu/~leemis/chart/UDR/UDR.html) contient encore plus de relations, ainsi que leur justifications mathématique. " ... ... @@ -610,18 +607,6 @@ "source": [ "![Relationships_among_some_of_univariate_probability_distributions](img/distributions.jpg)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { ... ... @@ -640,7 +625,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.5.0" "version": "3.6.5" } }, "nbformat": 4, ... ...
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 ... ... @@ -376,7 +376,7 @@ "step=(droite-gauche)/nb_batons\n", "weights=np.ones_like(X)/step/len(X)\n", "\n", "plt.hist(X,bins=nb_batons,weights=weights,range=[gauche,droite],edgecolor=\"k\")\n", "plt.hist(X,bins=bins,weights=weights,range=[gauche,droite],edgecolor=\"k\")\n", "x=np.linspace(gauche,droite,200)\n", "plt.plot(x, gaussian_density(x));" ] ... ...
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alea/book.tplx 0 → 100644
 % Default to the notebook output style ((* if not cell_style is defined *)) ((* set cell_style = 'style_ipython.tplx' *)) ((* endif *)) % Inherit from the specified cell style. ((* extends cell_style *)) %=============================================================================== % Latex Book %=============================================================================== ((* block predoc *)) ((( super() ))) ((* block tableofcontents *))\tableofcontents((* endblock tableofcontents *)) ((* endblock predoc *)) ((* block docclass *)) \documentclass[10pt, reprint, floatfix, groupaddress, prb, twoside]{book} \usepackage[francais]{babel} % Pour la langue française % Force floating figures to be [H] \usepackage{float} \floatplacement{figure}{H} \usepackage{amsmath} \usepackage{amssymb} \usepackage{mathrsfs} \usepackage{makeidx} \usepackage[svgnames]{xcolor} % Required for colour specification \definecolor{pythonblue}{rgb}{0.21,0.45,0.65} \definecolor{pythonyellow}{rgb}{1.,0.77,0.18} ((* endblock docclass *)) ((* block markdowncell scoped *)) ((( cell.source | citation2latex | strip_files_prefix | convert_pandoc('markdown', 'json',extra_args=[]) | resolve_references | convert_pandoc('json','latex', extra_args=["--top-level-division=chapter"]) ))) ((* endblock markdowncell *)) ((* block maketitle *)) \begin{titlepage} \vspace*{\baselineskip} \centering {\Large Matthieu Boileau \& Vincent Legoll} \vspace*{0.167\textheight} \raggedleft \textbf{\fontsize{30}{36}\selectfont Apprendre}\\[\baselineskip] \textbf{\textcolor{pythonblue}{\fontsize{50}{60}\selectfont Python}}\\[\baselineskip] \textcolor{pythonyellow}{\fontsize{30}{36}\selectfont \textit{pour les sciences}} \vspace*{0.1\textheight} \centering %\includegraphics{fig/head_of_python_molurus_frame} \vfill {\large CNRS - Université de Strasbourg} \end{titlepage} ((* endblock maketitle *)) ((* block bibliography *)) \bibliographystyle{unsrt} \bibliography{riemann} ((* endblock bibliography *)) ((*- block any_cell -*)) ((*- if not 'hide' in cell['metadata'].get('tags', {}) -*)) ((( super() ))) ((*- endif -*)) ((*- endblock any_cell -*)) ((* block input scoped *)) ((( add_my_prompt(cell.source | highlight_code(strip_verbatim=True), cell, 'In ', 'incolor') ))) ((* endblock input *)) % Purpose: Renders an output/input prompt ((* macro add_my_prompt(text, cell, prompt, prompt_color) -*)) ((*- if cell.execution_count is defined -*)) ((*- set execution_count = "" ~ (cell.execution_count | replace(None, " ")) -*)) ((*- else -*)) ((*- set execution_count = " " -*)) ((*- endif -*)) ((*- set indention = " " * (execution_count | length + 7) -*)) \begin{Verbatim}[fontsize=\small,commandchars=\\\{\}] ((( text | add_prompts(first='{\color{' ~ prompt_color ~ '}' ~ prompt ~ '[{\\color{' ~ prompt_color ~ '}' ~ execution_count ~ '}]:} ', cont=indention) ))) \end{Verbatim} ((*- endmacro *))
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 \relax \providecommand\zref@newlabel{} \@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}} \@writefile{toc}{\contentsline {section}{\numberline {2}Pr\IeC {\'e}liminaires sur la th\IeC {\'e}orie de la mesure}{1}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.1} Mesure et loi}{2}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.2}La notation avec $dx$}{2}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.3}Int\IeC {\'e}grale}{2}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.4}Mesures de Lebesgue}{2}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.5}L'indicatrice}{3}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.6}La Dirac}{3}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.7}Int\IeC {\'e}grale avec des Dirac}{4}} \@writefile{toc}{\contentsline {section}{\numberline {3}D\IeC {\'e}finition de la loi d'un objet al\IeC {\'e}atoire}{4}} \@writefile{toc}{\contentsline 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 \relax \providecommand\zref@newlabel{} \@writefile{toc}{\contentsline {section}{\numberline {1}Introduction : c'est quoi les stats ? }{1}} \@writefile{toc}{\contentsline {section}{\numberline {2}Convergence en loi}{2}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.1}Informatiquement}{2}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.2}D\IeC {\'e}finition}{2}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.3}Exemple \IeC {\a} avoir en t\IeC {\^e}te}{3}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.4}Caract\IeC {\'e}risation}{3}} \@writefile{toc}{\contentsline {section}{\numberline {3}Th\IeC {\'e}or\IeC {\e}me central limite}{3}} \@writefile{toc}{\contentsline {subsection}{\numberline {3.1}Enonc\IeC {\'e}}{3}} \@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Monte-Carlo}{4}} \@writefile{toc}{\contentsline {subsection}{\numberline {3.3}G\IeC {\'e}n\IeC {\'e}ralisation}{4}} \@writefile{toc}{\contentsline {subsection}{\numberline {3.4}Intervalle de confiance}{5}} \@writefile{toc}{\contentsline {section}{\numberline {4}Vecteur al\IeC {\'e}atoires}{5}} \@writefile{toc}{\contentsline {subsection}{\numberline {4.1}Vecteur al\IeC {\'e}atoire}{5}} \@writefile{toc}{\contentsline {subsection}{\numberline {4.2}Interpr\IeC {\'e}tation g\IeC {\'e}om\IeC {\'e}trique de la covariance}{6}} \@writefile{toc}{\contentsline {subsection}{\numberline {4.3}Exo}{6}} \newlabel{exoab}{{4.1}{6}} \@writefile{toc}{\contentsline {subsection}{\numberline {4.4}Dataframe}{7}} \@writefile{toc}{\contentsline {subsection}{\numberline {4.5}D\IeC {\'e}formation de la densit\IeC {\'e}}{7}} \@writefile{toc}{\contentsline {section}{\numberline {5}Vecteur gaussien}{7}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.1}gaussien standart $\to$ gaussien g\IeC {\'e}n\IeC {\'e}ral}{7}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.2} gaussien g\IeC {\'e}n\IeC {\'e}ral $\to$ gaussien standart}{7}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.3}Densit\IeC {\'e}}{8}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.4}Covariance nulle et ind\IeC {\'e}pendance}{8}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.5}TCL en dimension $p$}{8}} \@writefile{toc}{\contentsline {section}{\numberline {6}Loi des estimateurs de l'esp\IeC {\'e}rance et de la variance}{8}} \@writefile{toc}{\contentsline {subsection}{\numberline {6.1}Intro}{8}} \@writefile{toc}{\contentsline {subsection}{\numberline {6.2}Loi du Chi2}{9}} \@writefile{toc}{\contentsline {subsection}{\numberline {6.3}Projection}{9}} \@writefile{toc}{\contentsline {subsection}{\numberline {6.4}Th\IeC {\'e}or\IeC {\`e}me de cochrane}{9}} \@writefile{toc}{\contentsline {section}{\numberline {7}TCL sans connaitre la variance}{10}} \@writefile{toc}{\contentsline {subsection}{\numberline {7.1}D\IeC {\'e}finition de la loi $t$ de Student.}{10}} \@writefile{toc}{\contentsline {subsection}{\numberline {7.2}Une sorte de TCL}{10}} \@writefile{toc}{\contentsline {subsection}{\numberline {7.3}D\IeC {\'e}mo}{10}} \@writefile{toc}{\contentsline {subsection}{\numberline {7.4}Conclusion}{11}} \@writefile{toc}{\contentsline {section}{\numberline {8}Tests statistiques}{11}} \@writefile{toc}{\contentsline {subsection}{\numberline {8.1}Test avec statistique positive}{11}} \@writefile{toc}{\contentsline {subsection}{\numberline {8.2}$p$-valeur}{12}} \@writefile{toc}{\contentsline {subsection}{\numberline {8.3}Exemple : le test du $\chi _2$ de Pearson}{12}} \@writefile{toc}{\contentsline {subsection}{\numberline {8.4}Le test du $\chi _2$ de Pearson (suite)}{13}} \@writefile{toc}{\contentsline {subsection}{\numberline {8.5}Test avec une statistique sign\IeC {\'e}e}{13}} \@writefile{toc}{\contentsline {subsection}{\numberline {8.6}Exemple : le test de Student}{14}} \@writefile{toc}{\contentsline {subsection}{\numberline {8.7}vraie vie et grosse-donn\IeC {\'e}e}{14}}