Commit 0d44c69e authored by vincentvigon's avatar vincentvigon
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vect alea

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......@@ -14,6 +14,15 @@
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......@@ -3,5 +3,5 @@
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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 |&nbsp; &nbsp; &nbsp; &nbsp; simplifié &nbsp; &nbsp; &nbsp; &nbsp; |&nbsp; &nbsp; &nbsp; &nbsp; complète &nbsp; &nbsp; &nbsp; &nbsp; |\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": []
},
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......@@ -640,7 +625,7 @@
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......
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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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