{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<H1 style=\"font-size:50px\">9: Statistica Elementare con Numpy e Matplotlib</H1>\n",
    "\n",
    "<div style = \"width:95%; padding:5px 10px 5px;  margin:50px 0px 10px 10px; font-size:18px\">\n",
    "<UL>\n",
    "    <LI> Le funzioni statistiche fondamentali in Numpy\n",
    "    <LI> Come disegnare l'istogramma delle frequenze\n",
    "    <LI> Come e perchè generare numeri casuali secondo una distribuzione data\n",
    "</UL>\n",
    "</div>\n",
    "\n",
    "<hr>\n",
    "\n",
    "        \n",
    "## 9.1 Nozioni fondamentali\n",
    "La statistica ha un ruolo fondamentale in Fisica e in ogni campo scientifico a base sperimentale. Numpy fornisce tutti i metodi necessari per la manipolazione statistica dei dati. Affrontiamo questo argomento dopo aver introdotto Matplotlib perchè\n",
    "la rappresentazione grafica dei dati è uno strumento estremamente utile.\n",
    "\n",
    "<div style= 'padding:20px 20px 0px 150px;'>\n",
    "    <img src=\"../Humour/extrapolating1.png\" width=\"500\" align=\"center\"/>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dati in un array numpy."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "dati = np.array([1.95,1.96,1.9,1.9,1.84,1.81,2.06,1.99,1.93,1.97,2.02,1.92,1.95,1.88,1.87,2.03,1.85,2.08,1.96,1.81,\n",
    "                2.07,1.91,1.79,1.99,1.97,1.95,1.96,1.93,1.83,2.09,2.02,2.09,1.84,1.86,1.96,2.03,1.93,1.9,1.94,1.87,\n",
    "                1.97,1.91,1.87,1.81,2.06,2.02,1.96,1.81,1.93,2.03,1.92,1.96,1.8,1.95,1.9,2.02,2.03,1.9,2.03,2.02,\n",
    "                1.96,1.9,1.98,1.87,1.9,1.89,1.84,2.06,1.93,2.06,1.93,1.93,1.9,1.9,1.9,1.93,1.86,1.83,1.96,1.81,2.03,\n",
    "                1.98,1.84,1.86,1.96,1.81,1.98,1.84,1.86,1.96,1.92,1.96,1.85,2.04,2,1.92,1.9,2.15,1.94,1.92])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "100"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "num_elementi = dati.size\n",
    "num_elementi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dati al quadrato"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dati_sq = dati*dati"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Il valor medio o media di un insieme di dati $x = [x_1,\\cdots,x_n]$ è\n",
    "$$ <x> = \\frac{\\sum_{i=1}^n x_i}{n} $$\n",
    "Media utilizzando solo la funzione sum"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "media1 = dati.sum()/num_elementi\n",
    "media1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Media utilizzando la funzione mean di numpy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.9357"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "media2 = dati.mean()\n",
    "media2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La varianza di un insieme di dati $x = [x_1,\\cdots,x_n]$ è\n",
    "$$ \\sigma^2  = \\,\\, <(x - <x>)^2> \\,\\, = \\frac{\\sum_{i=1}^n (x_i-<x>)^2}{n} = \\,\\,<x^2> -<x>^2$$\n",
    "Varianza calcolata espicitamente"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "varianza1 = (dati_sq - 2.*media1*dati + media1*media1).sum()/num_elementi # Notice array + const*array + const\n",
    "varianza1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Varianza calcolata usando la funzione var di numpy."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "varianza2 = dati.var()\n",
    "varianza2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Deviazione standard, $\\sigma = \\sqrt{\\sigma^2}$, calcolata dalla varianza e usando la funzione std di numpy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "deviazione_std1 = np.sqrt(varianza2)\n",
    "deviazione_std1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.07747586721037715"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "deviazione_std2 = dati.std()\n",
    "deviazione_std2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Talvolta è utile estrarre i dati a meno di un numero dato di deviazioni standard dal valor medio, oppure quelli che distano più\n",
    "di un numero dato di deviazioni standard dal valor medio. Nell'esempio selezioniamo i dati all'interno di una deviazione standard:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dati1 = np.array([n for n in dati if np.absolute(n - media1) < deviazione_std1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dati1.size"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div style = \"width:95%; padding:5px 10px 5px; border-style:solid; border-width:0px 0px 0px 10px; border-color:#b8d0d0; margin:20px 0px 10px 10px\">\n",
    "    <h4> Imparare Facendo</h4>\n",
    "    \n",
    "Dato l'array\n",
    "<code>\n",
    "my_arr = np.array([3.04645601, 2.97244915, 3.11895648, 2.23631771, 2.83300643,\n",
    "       2.05404508, 2.75985706, 2.26921367, 1.37480605, 2.02558085,\n",
    "       2.03489553, 2.9879711 , 1.96904577, 2.26633488, 2.25061096,\n",
    "       2.19332838, 2.01392679, 3.11555729, 1.86606049, 3.05021054,\n",
    "       3.03353987, 2.31818007, 2.79232123, 3.33861491, 3.28415856,\n",
    "       1.99223361, 2.84573136, 1.79728384, 3.02507785, 1.66469195,\n",
    "       1.9154713 , 2.43314196, 1.86340421, 1.90131182, 2.09963155,\n",
    "       2.12451288, 1.77265763, 1.81662815, 2.0122717 , 2.82934715,\n",
    "       1.72616883, 2.86297194, 2.9404613 , 2.96369557, 1.8376963 ,\n",
    "       1.81889892, 2.08233386, 1.86941276, 1.83248482, 3.12861456,\n",
    "       1.54804543, 2.89724744, 2.97500892, 2.28660094, 3.51159172,\n",
    "       1.64804177, 2.9250396 , 2.98698285, 2.17577323, 2.43331005,\n",
    "       2.03573614, 2.96237528, 3.14320927, 2.13393559, 2.55083613,\n",
    "       1.72551903, 1.56344938, 3.31847721, 3.19368425, 2.81418586,\n",
    "       2.79420806, 2.88605616, 2.08231959, 1.68927766, 1.89277468,\n",
    "       2.03634711, 3.01241034, 1.95824444, 1.84229893, 1.79093756,\n",
    "       2.16777509, 1.91055935, 2.16076815, 1.99936357, 2.42660732,\n",
    "       2.09021026, 3.25206981, 2.55240002, 2.83482414, 2.0150959 ,\n",
    "       3.31566771, 3.58996448, 2.58405186, 2.74445492, 2.80650089,\n",
    "       1.25237511, 2.04484102, 1.95878434, 1.99383903, 1.63776293,\n",
    "       3.08590679, 3.42000501, 1.69301131, 1.45661319, 2.75442641,\n",
    "       3.01573607, 2.64220989, 2.37487723, 1.83034393, 1.74794294,\n",
    "       1.92414741, 2.91175392, 3.25243102, 2.91543309, 3.22180813,\n",
    "       1.89410574, 1.83451938, 1.60409685, 3.24371334, 2.04293352,\n",
    "       2.89538543, 2.05009924, 2.20696778, 2.05073664, 1.83387137,\n",
    "       2.07646022, 3.34614149, 3.30588549, 3.25491247, 1.55441846,\n",
    "       2.19207954, 2.11273179, 2.92539792, 3.19288315, 1.2374957 ,\n",
    "       2.17409141, 1.76835303, 2.12725474, 2.91318578, 2.96264334,\n",
    "       2.18750678, 1.96060764, 3.41421698, 2.78839075, 1.56933989,\n",
    "       1.64822396, 1.8383093 , 1.49851104, 3.03809049, 2.84892792,\n",
    "       1.6809436 , 3.12957714, 3.01568747, 1.87809285, 1.71435392,\n",
    "       1.83658257, 1.94510093, 3.04703205, 2.97698006, 1.99492519,\n",
    "       3.07521061, 1.96260647, 2.91868837, 1.94448569, 2.96816534,\n",
    "       2.90343856, 2.33124355, 1.88310601, 1.93171459, 2.54151343])\n",
    "</code>\n",
    "calcolate media e standard deviation.\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 9.2 Istogramma delle frequenze\n",
    "\n",
    "L'istogramma delle frequenze è sovente il modo migliore per un primo esame dei dati"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "min = dati.min()\n",
    "min"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "max = dati.max()\n",
    "max"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La conoscenza del minimo e del massimo valore dei dati è utile per determinare il `range` dei bin. Il numero di bin deve essere adattato di volta in volta per aiutare l'analisi."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "nbins = 10\n",
    "xrange = (1.75,2.20)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nell'esempio seguente vengono inserite tre linee verticali per segnalare il valor medio e i valori della variabile a più e meno una deviazione standard dal valor medio. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "nevent, bins, patches = ax.hist(dati, nbins, range=xrange)\n",
    "ax.plot(np.ones(2)*media2,[0,nevent.max()+1],label=\"media\")\n",
    "ax.plot(np.ones(2)*media2-deviazione_std2,[0,nevent.max()+1],label=\"media - $\\sigma$\")\n",
    "ax.plot(np.ones(2)*media2+deviazione_std2,[0,nevent.max()+1],label=\"media + $\\sigma$\")\n",
    "ax.legend();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "nevent       # Numero di eventi in ciacun bin"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "bins         # Estremi dei bin"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I patches sono i rettangoli (blu in questo caso) che vengono usati per disegnare l'istogramma."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Il metodo `hist` ha un parametro booleano `density` con default `False`. Se `density=True` ogni bin mostra il numero di eventi nel bin diviso per il numero totale di eventi e diviso per la larghezza del bin in modo che l'area sotto l'istogramma sia 1. Utile per confontare la densit&agrave; dei dati con una densit&agrave; di\n",
    "probabilit&agrave; analitica, per esempio gaussiana."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "nevent, bins, patches = ax.hist(dati, nbins, range=xrange, density=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div style = \"width:95%; padding:5px 10px 5px; border-style:solid; border-width:0px 0px 0px 10px; border-color:#b8d0d0; margin:20px 0px 10px 10px\">\n",
    "    <h4> Imparare Facendo</h4>\n",
    "    Fate il diagramma delle frequenze dell'array <b>my_arr</b>. In particolare\n",
    "<OL>\n",
    "<LI>Determinate un <b>range</b> ragionevole. \n",
    "<LI>Variate il numero di bin, notando come un sia un numero troppo piccolo che uno troppo grande mascherano l'andamento dei dati.\n",
    "<LI>Verificate se il valor medio e la standard deviation sono parametri utili nella comprensione di questo set di dati\n",
    "</OL>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div style = \"width:95%; padding:5px 10px 5px; border-style:solid; border-width:0px 0px 0px 10px; border-color:#ff0000; margin:20px 0px 10px 10px\">\n",
    "    <h4> Attenzione!</h4>\n",
    "    Lo studio grafico dei dati non va mai trascurato.<BR>\n",
    " In un notebook a parte <A HREF=\"anscombe.ipynb\"> anscombe.ipynb</A> potete trovare un classico esempio di quanto sia pericoloso affidarsi completamente ai parametri statisti fondamentali: quattro set di dati che hanno la stessa media, la stessa deviazione standard e vengono interpolati dalla stessa retta pur essendo completamente diversi.\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div style= 'padding:20px 20px 0px 70px;'>\n",
    "    <img src=\"../Humour/extrapolating2.png\" width=\"850\" align=\"left\"/>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 9.3 Numeri Casuali\n",
    "\n",
    "Nelle versioni più recenti di Numpy (>1.17) è stata introdotto un oggetto `default_rng` i cui metodi possono essere utilizzati \n",
    "per generare set di valori distribuiti secondo le singole distribuzioni di probabilità. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "from numpy.random import default_rng\n",
    "\n",
    "rng = default_rng()\n",
    "\n",
    "#help(np.random.default_rng)\n",
    "#help(rng)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Come generare numeri distribuiti secondo la distribuzione normale standard, con valor medio $\\mu = 0.0$, e deviazione standard $\\sigma = 1.0\\,$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m1 = rng.normal(size=2000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "nbins = 30\n",
    "xrange = (-5,5)    # ntupla\n",
    "fig, ax = plt.subplots()\n",
    "nevent, bins, patches = ax.hist(m1, nbins, range=xrange)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Come generare numeri distribuiti secondo la distribuzione normale con $\\mu = -2.0,\\, \\sigma = 0.3\\,$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m2 = rng.normal(loc=-2., scale=0.3, size=2000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "nbins = 300\n",
    "xrange = (-5,1)    # ntupla\n",
    "fig, ax = plt.subplots()\n",
    "nevent, bins, patches = ax.hist(m2, nbins, range=xrange)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Un modo di generare numeri **reali** secondo la distribuzione uniforme (tutti i punti sono equiprobabili) standard (L'intervallo di definizione &egrave; $[0,1]$). "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#help(rng.uniform)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m3 = rng.uniform(size=2000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "nbins = 12\n",
    "xrange = (-0.1,1.1)    # ntupla\n",
    "fig, ax = plt.subplots()\n",
    "nevent, bins, patches = ax.hist(m3, nbins, range=xrange)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Un modo di generare numeri **interi** uniformemente distribuiti fra un minimo (incluso) e un massimo (escluso). "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#help(rng.integers)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m4 = rng.integers(0,high=100,size=20)\n",
    "m4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "    In letteratura si trovano ancora i generatori di numeri casuali meno recenti ( sempre in numpy.random)\n",
    "    ======================================================================================================\n",
    "    rand                 Uniformly distributed values.\n",
    "    randn                Normally distributed values.\n",
    "    ranf                 Uniformly distributed floating point numbers.\n",
    "    randint              Uniformly distributed integers in a given range.\n",
    "    ======================================================================================================\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Si può fissare il \"seme\" del generatore di numeri casuali in modo da ottenere la stessa sequenza più volte.\n",
    "Utile qundo si vogliono capire le analisi statistiche fatte da qualcun altro."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "rng1 = default_rng(12345)\n",
    "rng2 = default_rng(12345)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m3_1 = rng1.uniform(size=200)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m3_2 = rng2.uniform(size=200)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m3_1 == m3_2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "rng3 = default_rng(12345)\n",
    "rng4 = default_rng(12345)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m1_1 =rng3.normal(size=1000)\n",
    "#m1_1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m1_2 =rng4.normal(size=1000)\n",
    "#m1_2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "all(m1_1 == m1_2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div style= 'padding:20px 20px 0px 150px;'>\n",
    "    <img src=\"../Humour/NormalDistribution.jpg\" width=\"400\" align=\"left\"/>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div style = \"width:95%; padding:5px 10px 5px; border-style:solid; border-width:0px 0px 0px 10px; border-color:#b8d0d0; margin:20px 0px 10px 10px\">\n",
    "    <h4> Imparare Facendo</h4>\n",
    "    \n",
    "Producete campioni casuali estratti da una distribuzione normale, con valor medio $\\mu = 3.$ e deviazione standard $\\sigma = 0.7$, di 50, 100, 500, 5000 elementi.\n",
    "    Di ciascun campione\n",
    "<OL>\n",
    "<LI>Fate l'istogramma delle frequenze.\n",
    "<LI>Sovrapponete all'istogramma delle frequenze la funzione densità di probabilità gaussiana di valor medio $\\mu$ e deviazione standard $\\sigma$, facendo attenzione che la curva e l'istogramma siano confrontabili. Cosa è necessario fare?\n",
    "</OL>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 9.4 Distribuzioni di probabilità\n",
    "\n",
    "Il modulo `scipy.stats` contiene le principali distribuzioni di probabilità , sia discrete che continue, funzioni che permettono di calcolare i parametri statistici più comuni di un set di dati, funzioni che eseguono test statistici e test di correlazione fra uno o più set di dati.<BR>\n",
    "Per ulteriori informazioni https://docs.scipy.org/doc/scipy/reference/stats.html."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 9.4.1 Distribuzione normale: $\\,\\,N(x) = \\frac{\\exp (-x^2/2)}{\\sqrt{2\\,\\pi}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import scipy.stats as stats\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Grafico della distribuzione"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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0ql6hRa8skl9ey0cZZ7g7JZYAXx1S2ddCAny5Y0Y07x7SoZZKi15Z5LU9eXiJcM/sWKujuK3lc+JobDZsSNWhlp5Oi171uwsNzWzYl8/CCTqk0pFGRgQzf2wEr+3No7FZh1p6Mi161e82HSzk/IVGfUCqH9w/J44zlfVsPXrG6ijKQlr0ql+1DKnMIzEyhJSEwVbHcXtXjhtKzOABvLI71+ooykJa9Kpfpeed44vTlSyfE69DKvuBt5dw3+w4UnPKyThdaXUcZREtetWvXtmdR0iAD7dMG2F1FI9xZ3IM/j5erN+dZ3UUZREtetVvSirreP/wae5MjiHQz8fqOB4jLNCPW6ZG8df9hZyvbbQ6jrKAFr3qN39KPUWTzXDfbP0Qtr8tnxvHhcZm3krXBcQ9kV1FLyILReS4iGSKyBMdbL9HRA61/tklIlPabMsVkcMickBEdDURD9XYOnXuFWMjiA8PsjqOx5kwYiAz4gbx2p48XUDcA3VZ9CLiTcvygDcAScAyEUlqt1sOcIUxZjLwE2Btu+0LjDFTO1v9RLm/LUeLKamq1yGVFlo+J47cslo+PVlqdRTVz+y5ok8BMo0x2caYBmADsKTtDsaYXcaYc60v9wDRfRtTubr1u/OIHjSAK8cNtTqKx7ph4nDCg/15VT+U9Tj2FH0U0PbGXkHre51ZCbzf5rUBtopIuois6uwgEVklImkiklZaqlcc7uRYcSWpOeXcOzsOb1342zJ+Pl4sS4nh78dLyC+vtTqO6kf2FH1HP5kd3uQTkQW0FP132rw9zxgznZZbPw+LyPyOjjXGrDXGJBtjkiMiIuyIpVzFq7vz8Pfx4q7kGKujeLy7Z8XiJcJre/Wq3pPYU/QFQNuf0GigqP1OIjIZeB5YYowpu/i+Maao9b8lwDu03ApSHqKyrpF39hdy85QRDAryszqOxxs+cADXjh/Gxn351DU2Wx1H9RN7in4fMEZEEkTED1gKbGq7g4jEAn8B7jPGnGjzfpCIhFz8O3AdcKSvwivn95f0AmobmvVDWCeyfG4c52obeffgv12vKTfVZdEbY5qAR4AtQAaw0RhzVERWi8jq1t1+CAwBnmk3jHIYsFNEDgKpwGZjzAd9fhbKKRljWL8njykxYUyODrM6jmo1Z+QQxgwN5tU9evvGU9j1eKIx5j3gvXbvPdvm7w8CD3ZwXDYwpf37yjN8lllGdmkNv/mS/hNwJiLCfXPi+OHfjnIgv4KpMWFWR1IOpk/GKodZv7tlqcAbdalAp3PrtCiC/LxZr7NaegQteuUQhRUX+CjjDHfNjNGlAp1QSIAvt02P5v8OnaasWpcadHda9MohXm+9/3vPLF0q0FktnxNHQ5ONN9N0/ht3p0Wv+lx9UzNv7svnqsRhRA8KtDqO6sSYYSHMGTmE1/ecolnnv3FrWvSqz713+DRlNQ3cP1eHVDq75XPiKKy4wN+PlVgdRTmQFr3qc+t35zEyPIh5o8KtjqK6cG3SMCJDA/RDWTenRa/61KGCCvafquC+OXF46bw2Ts/H24t7ZsWy4+RZskqrrY6jHESLXvWp9bvzCPTz5vYZOoGpq1iaEouvt+islm5Mi171mfKaBjYdLOK26VGEBvhaHUfZKSLEnxsnDeft9AKq65usjqMcQIte9Zk39+XT0GRj+Zx4q6Ooblo+N56q+ibe2V9odRTlAFr0qk802wyv7cljzsghjB0WYnUc1U3TYsKYFDWQ9btyMUaHWrobLXrVJ/5+rITCigs6pNJFiQjL58RxsqSa3dllXR+gXIoWveoTr+zKZfjAAK4ZP8zqKKqHbp4ygkGBvryyK9fqKKqPadGrXjt5poqdmWe5d3YcPt76T8pVBfh6szQllg+/OEPBOV1q0J3oT6XqtVd257auR6rz2ri6e2e33HrTuerdixa96pXzFxr5y+eFLJ4ygsG6VKDLiwobwPUTInlTlxp0K3YVvYgsFJHjIpIpIk90sP0eETnU+meXiEyx91jl2t5Ky6e2oZkH5sZbHUX1kfvnxlNR28jfDuhQS3fRZdGLiDfwNHADkAQsE5GkdrvlAFcYYyYDPwHWduNY5aKabYb1u/NIjhvExKiBVsdRfWRWwmASI0N46TMdauku7LmiTwEyjTHZxpgGYAOwpO0OxphdxphzrS/3ANH2Hqtc17bjJZwqr+V+vZp3KyLCA3PjOVZcRWpOudVxVB+wp+ijgLYrExS0vteZlcD73T1WRFaJSJqIpJWWltoRS1nt5V25DAv1Z+HESKujqD62ZGoUYYG+vKxDLd2CPUXf0RSEHf4+JyILaCn673T3WGPMWmNMsjEmOSIiwo5YykonzlSx4+RZls+Jx1eHVLqdAX7eLJ0Zy5ajxTrU0g3Y8xNaAMS0eR0NFLXfSUQmA88DS4wxZd05Vrmelz7LxV+HVLq15XPiEBHW66yWLs+eot8HjBGRBBHxA5YCm9ruICKxwF+A+4wxJ7pzrHI952oaeGd/AbdOi9IhlW5sRNgAbpgYyRupp6jRWS1dWpdFb4xpAh4BtgAZwEZjzFERWS0iq1t3+yEwBHhGRA6ISNqljnXAeah+9Ma+U9Q12lgxL8HqKMrBVsxLoKquibc/L7A6iuoFccbhU8nJySYtLc3qGKoDjc02Lv/FJ4weGsxrD86yOo5yMGMMtzyzi6oLjXz0zSt01TAnJiLpxpjkjrbpp2iqWz44UkxxZR0r5sVbHUX1AxHhy/PiyT5bw/YTOhrOVWnRK7sZY3hhZw7xQwJZMG6o1XFUP1k0aTjDQv15YWeO1VFUD2nRK7ul553jQH4FX74sQX+F9yC+3l4snxPPzsyzZJyutDqO6gEtemW353fkMHCAL3fowt8e555ZsQzw9eb5HXpV74q06JVd8spq2PJFMffOjiXQz8fqOKqfhQX6cWdyNJsOFnKmss7qOKqbtOiVXV7cmYOPl+jC3x5sxbwEmmyG9btzrY6iukmLXnWporaBjWkFLJ4SxbDQAKvjKIvEhwdxXdIwXttzitoGfYDKlWjRqy69vvcUFxqbefByfUDK033l8pGcv9DIn9P1ASpXokWvLqm+qZlXduVy2ehwxg8PtTqOstiMuEFMiQnjhZ05NNuc72FL1TEtenVJf91fSElVPavmj7Q6inICIsJX548kr6yWD44UWx1H2UmLXnXKZjM892k2ScNDuXxMuNVxlJO4fkIk8UMCeXZ7lq5A5SK06FWnPso4Q3ZpDV+9YiQi+oCUauHtJXxl/kgOF55nd1ZZ1wcoy2nRq04992k20YMGcOOk4VZHUU7m9unRhAf78eyn2VZHUXbQolcd2pdbTnreOb5y+Uh8dAUp1U6Arzcr5iXw6YlSjhadtzqO6oL+BKsOPbc9i8FBftyZHNP1zsoj3TsrjiA/b57brlf1zk6LXv2bY8WVfJRRwv1z4hng5211HOWkBgb6cvesWDYfPs2pMl1X1pnZVfQislBEjotIpog80cH2RBHZLSL1IvJ4u225InK47cpTyrk9/UkWQX7e3D83zuooysk9ePlIvEVYsz3L6ijqErosehHxBp4GbgCSgGUiktRut3LgUeDXnXyZBcaYqZ2tfqKcR87ZGjYfKuK+OfGEBep6sOrShoUGcOfMaN5OL6D4vE525qzsuaJPATKNMdnGmAZgA7Ck7Q7GmBJjzD6g0QEZVT9asy0TX28vVl6m0x0o+3x1/iiajWGtjsBxWvYUfRSQ3+Z1Qet79jLAVhFJF5FVne0kIqtEJE1E0kpLdckyKxRWXOAvnxeyLCWWiBB/q+MoFxEzOJBbpkbxp9Q8yqrrrY6jOmBP0Xf0pEx3HoebZ4yZTsutn4dFZH5HOxlj1hpjko0xyREREd348qqvrN2ehQg63YHqtq8tGEV9k40XP9OFSZyRPUVfALQdYxcNFNn7DYwxRa3/LQHeoeVWkHIyJZV1bNiXz23TohkRNsDqOMrFjIoIZtHE4azflUdFbYPVcVQ79hT9PmCMiCSIiB+wFNhkzxcXkSARCbn4d+A64EhPwyrHWbM9iyab4eEFo62OolzU168eTVV9Ey/qIuJOp8uiN8Y0AY8AW4AMYKMx5qiIrBaR1QAiEikiBcA3ge+LSIGIhALDgJ0ichBIBTYbYz5w1MmonjlTWcfre09x+/QoYocEWh1HuajEyFAWTYrkxc9y9areydi1+Kcx5j3gvXbvPdvm78W03NJprxKY0puAyvHWbMvCZjM8smCM1VGUi/vG1WN573Axz+/I4fHrx1kdR7XSJ2M9XPH5Ov6Ueorbp0fr1bzqtXGRIdw4aTgv78rlXI1e1TsLLXoPt2ZbZsvV/FV6b171jUevHkNNQxPP79Rx9c5Ci96DFVVc4I3UfO6YEU3MYL2aV31jXGQIiyYN5+XPcnVcvZPQovdgv//oJIBezas+9x/XjOFCYzPPbNM5cJyBFr2Hyiqt5q30fO6eFUv0IL2aV31r9NAQbpsezat78iiquGB1HI+nRe+hfrv1BAG+3no1rxzmsWvGgPnnb47KOlr0HuhI4Xk2Hz7NyssSCA/WOW2UY0QPCuTuWbG8lZ5PVmm11XE8mha9B/rlluOEBfryFZ3TRjnYI1eNJsDXm99uPWF1FI+mRe9hdmWd5dMTpXztylGEBvhaHUe5ufBgfx68LIHNh09zML/C6jgeS4veg9hshqc2ZxAVNoDlc+KtjqM8xFfmjyQ82I+nNmdgTHcmvlV9RYveg/xlfyFHiyr59sJxBPjqWrCqf4QE+PLYNWNJzS1ny9EzVsfxSFr0HuJCQzO/3nKcydEDuXnyCKvjKA+zdGYMo4cG8/P3M2hoslkdx+No0XuI53dkU1xZx/cWjcfLq6O1ZJRyHB9vL767KJHcslpe35tndRyPo0XvAUqq6nh2exbXJQ1j1sghVsdRHmrBuKHMGz2E3398kvO1urx0f9Ki9wC/eP84Dc02/nPReKujKA8mInxvURKVFxr5n490uGV/0qJ3c+l553j78wJWXjaShPAgq+MoD5c0IpR7ZsWxfncuGacrrY7jMewqehFZKCLHRSRTRJ7oYHuiiOwWkXoRebw7xyrHabYZntx0hGGh/nxdpzpQTuL/XTeWgQN8eXLTUR1u2U+6LHoR8QaeBm4AkoBlIpLUbrdy4FHg1z04VjnIm/vyOVJYyXcXjSfI367FxJRyuLBAPx6/fhypOeW8e+i01XE8gj1X9ClApjEm2xjTAGwAlrTdwRhTYozZB7T/hKXLY5VjVNQ28Kstx0hJGMziKTqcUjmXpTNjmRgVyn9vzqCmvsnqOG7PnqKPAvLbvC5ofc8edh8rIqtEJE1E0kpLS+388qozP3//GJV1Tfx48QREdDilci7eXsKPF0+kuLKO//lQP5h1NHuKvqOWsPfGmt3HGmPWGmOSjTHJERERdn551ZG92WVs2JfPg5clMH54qNVxlOrQjLhB3D0rlhc/y+FwwXmr47g1e4q+AIhp8zoaKLLz6/fmWNUD9U3N/Oc7h4keNIBvXDPG6jhKXdJ3FiYyJNif/3znEE3N+sSso9hT9PuAMSKSICJ+wFJgk51fvzfHqh5Ysy2L7NIafnrLRAL99ANY5dwGDvDlRzdP4EhhJS/vyrU6jtvqsuiNMU3AI8AWIAPYaIw5KiKrRWQ1gIhEikgB8E3g+yJSICKhnR3rqJPxdJklVTzzSRaLp4zgynFDrY6jlF0WTYrk6sSh/GbrCfLLa62O45bEGcexJicnm7S0NKtjuJSmZht3PLub3LIaPvyPK4gI0ZWjlOsorLjAdb/dztTYMF798iydj6kHRCTdGJPc0TZ9MtZNrN2RzYH8Cv5ryUQteeVyosIG8P2bkvgss0wnPXMALXo3cLy4it99eJJFkyK5efJwq+Mo1SNLZ8ZwxdgI/vu9Y+SV1Vgdx61o0bu4xmYb39x4gJAAH36yZKKOmVcuS0T4+e2T8PEWvvXWIWw257ut7Kq06F3cHz4+ydGiSv77tkkMCdZbNsq1DR84gB/dPIHU3HLW7ci2Oo7b0KJ3YbuzyvjjJ5ncMSOa6ydEWh1HqT5x2/QobpgYya+2HNcFxfuIFr2LKq9p4LE395MQHsSPF0+wOo5SfUZE+PltkxkWGsDX39hPVZ0uUtJbWvQuyBjDt/98kHM1jfxh6TSdmVK5nYGBvvx+6VQKztXy/b8e0emMe0mL3gW99FkuH2WU8MQNiUyMGmh1HKUcIjl+MI9dM5a/HSjirfQCq+O4NC16F5OaU85/v5fBNeOHsWJevNVxlHKohxeMZu6oIXz/r0c4UqgTn/WUFr0LOVNZx9de/5yYwYH89q4pOpRSuT1vL+F/l00jPMiPr76aTnlNg9WRXJIWvYtoaLLx0Gvp1DY08dx9MwgN8LU6klL9YkiwP8/eN4PS6noefWM/zTq+vtu06F2AMYYnNx3l81MV/OqOKYwdFmJ1JKX61eToMH66ZCI7M8/y8/czrI7jcnS4hgtYtyObN1JP8dCVo7hRpzhQHurOmTEcKTrPuh05JIQHc/esWKsjuQwteif3wZFifvb+MW6cNJxvXTfO6jhKWeqHNyVxqryWH/ztCNGDBjB/rK5GZw+9dePEDuZX8Nib+5kaE8Zv7pyiU7cqj+fj7cUf757OmKHBPPz65xwvrrI6kkvQondSmSXVrHh5H+HB/qxbnkyAr7fVkZRyCsH+Prz4wEwG+Hmz/MW9uliJHewqehFZKCLHRSRTRJ7oYLuIyB9atx8SkelttuWKyGEROSAiupqIHQrO1XLfC3vxEuHVlbMI18nKlPoXI8IGsH5lCnWNNu59YS8llXVWR3JqXRa9iHgDTwM3AEnAMhFJarfbDcCY1j+rgDXtti8wxkztbPUT9U+lVfXc+/xeauqbeHVlCgnhQVZHUsopJUaG8tKKmZRW1XPfC6lU1OoY+87Yc0WfAmQaY7KNMQ3ABmBJu32WAOtNiz1AmIjo8JBuuljyZyrreWlFCuOHh1odSSmnNj12EOuWJ5NztkbL/hLsKfooIL/N64LW9+zdxwBbRSRdRFZ19k1EZJWIpIlIWmlpqR2x3Evx+TruWrubU+W1vHB/MjPiBlkdSSmXMG90OM/eN53jxVUsW7eXsup6qyM5HXuKvqOhHu0fTbvUPvOMMdNpub3zsIjM7+ibGGPWGmOSjTHJERGeNWSqsOICd63dzZnzdbzy5RTmjg63OpJSLuWqxGE8f38y2aXVLFu3h5IqvWfflj1FXwDEtHkdDRTZu48x5uJ/S4B3aLkVpFqdOFPFl9bsorymgVcfnEVKwmCrIynlkuaPjeClFTPJL7/Anc/u1nVn27Cn6PcBY0QkQUT8gKXApnb7bAKWt46+mQ2cN8acFpEgEQkBEJEg4DrgSB/md2l7ssu4Y80uGm2GN74ym+mxertGqd6YOyqc1x6cRcWFRm57ZhcHdIUqwI6iN8Y0AY8AW4AMYKMx5qiIrBaR1a27vQdkA5nAOuBrre8PA3aKyEEgFdhsjPmgj8/BJW06WMTyF1KJCPHnLw/N1XnlleojM+IG8fZDcwn092bp2t18+MUZqyNZTpxx5Zbk5GSTluaeQ+6bbYbfbD3OM9uymBnfMmIgLNDP6lhKuZ3SqnpWvrKPw4Xn+X/XjuVrV45266fLRSS9syHs+mRsP6qobWDFy/t4ZlsWy1Jiee3BWVrySjlIRIg/b66aw5IpI/j11hM89Ho61fVNVseyhBZ9P9l/6hw3/3Ene7LK+Nltk/jZbZPw99FpDZRypAF+3vzPXVP5wU1JfJRRwuI/7uRokeetVKVF72DNNsP/fnySO57djc0GG746m2UpOr2qUv1FRFh5WQKvrZxFTX0Ttz69i+d3ZGPzoAVMtOgdKOdsDcvW7uE3H57gpsnDef+xy3VkjVIWmTNqCB98Yz5Xjovgp5szWP5iKgXnPGNCNP0w1gEam22s25HN7z46ib+PFz9ePIHbpkdbHUspRcuKbW+k5vPU5i8wwOPXjeP+ufF4u/gHtZf6MFaLvo/tzS7jR+9+QcbpSm6YGMmPF09gaGiA1bGUUu0UVlzge+8cZtvxUqZED+RHiycwzYV/49ai7wenymr52fsZvH+kmOEDA3jy5gksnBhpdSyl1CUYY9h0sIifbs6gtKqeW6aO4NsLExkRNsDqaN12qaLXpQR76UxlHc98kskbqfl4ewnfvHYsX7l8JAP8dESNUs5ORFgyNYqrxw9jzbZM1u3I4YOjxdw3O46vXjHKbdaC0Cv6Hjp9/gLrPs3htb15NNsMd0yP5rFrxzB8oOtdCSilWuSX1/LbD0/wtwOF+Pt4s3xuHCvnJbjE7Ve9ddOHjhSeZ92ObDYfOo0BbpsWxdevGkPskECroyml+khWaTV/+Pgkmw4W4evlxeKpI3jw8gQSI513jQgt+l6qqW/i/w4V8UZqPgfyKwj29+GumTE8MDeemMFa8Eq5q5yzNbz0WQ5vpRVwobGZlPjB3DUzhkWThjvd7Vkt+h5obLaxM/Ms7x4sYsuRYmoamhk9NJilM2O4c2YMoQG+luZTSvWfitoGNuzLZ0PqKXLLagkJ8GHRxOHcPGUEs0cOxsfb+keStOjtVFPfxI6TZ/k44wwfZZzhXG0jIQE+LJwQyV0zY5gRNwgR1x5rq5TqOWMMe7LL2ZiWz9ajLReA4cH+XJs0lKsThzFvdLhlV/o66qYTTc02jhRV8lnmWXZlnWVfzjkamm2EBPhwVeJQbpo8gvljw3VOGqUU0DJKZ86oIcwZNYS6xmY+OVbC/x06zbsHT/NGaj7+Pl6kJAxm3uhw5o0KJ2lEqFM8iOUxV/TGGM5U1nOk8DwHCypIzzvHgfwKahuaAUiMDOGy0eFcPX4YyfGD8HWCX8WUUq6hocnG3pwyPs4o4bPMs5wsqQYg2N+HabFhTI8dxNSYMCZEhTI0xDEjeDzq1o0xhrKaBnLP1pBZUs2JM9WcLKki43QlZ6tbVoj39hLGDw9hRuwgZsQPZu6oIW4zXlYpZb2Syjp2ZZWRlldOel4Fx4sruTiH2tAQfxKHhzJ2aDBjh4UwamgwCeFBDAr07dWt4V4XvYgsBH4PeAPPG2N+3m67tG5fBNQCDxhjPrfn2I70pOibbYZbn/mMnNIaqtrMOR3g68WYoSGMiwxh4ohQJkYNJGlEKIF+Hn3XSinVj6rrm/iiqJIjhec5UnieY8VVZJVWU99k+8c+oQE+JEaG8uZXZ/eo8Ht1j15EvIGngWtpWQR8n4hsMsZ80Wa3G4AxrX9mAWuAWXYe2ye8vYSR4UFMiwkjPjyI+CFBjIoIJnrQALdeVUYp5fyC/X1ISRhMSsLgf7zXbDPkl9eSfbaanLO15JXV0Nhsc8iAD3sua1OATGNMNoCIbACWAG3Legmw3rT8erBHRMJEZDgQb8exfeZ3S6c54ssqpVSf8/aSlovS8CCHfy97PnGMAvLbvC5ofc+efew5FgARWSUiaSKSVlpaakcspZRS9rCn6Dv6PaL9jf3O9rHn2JY3jVlrjEk2xiRHRETYEUsppZQ97Ll1UwDEtHkdDRTZuY+fHccqpZRyIHuu6PcBY0QkQUT8gKXApnb7bAKWS4vZwHljzGk7j1VKKeVAXV7RG2OaROQRYAstQyRfNMYcFZHVrdufBd6jZWhlJi3DK1dc6liHnIlSSqkOud0DU0op5YkuNY5en/NXSik3p0WvlFJuzilv3YhIKZDXw8PDgbN9GMdK7nIu7nIeoOfijNzlPKB35xJnjOlwbLpTFn1viEhaZ/epXI27nIu7nAfouTgjdzkPcNy56K0bpZRyc1r0Sinl5tyx6NdaHaAPucu5uMt5gJ6LM3KX8wAHnYvb3aNXSin1r9zxil4ppVQbWvRKKeXm3LLoReQnInJIRA6IyFYRGWF1pp4QkV+JyLHWc3lHRMKsztRTIvIlETkqIjYRcbmhcCKyUESOi0imiDxhdZ7eEJEXRaRERI5YnaU3RCRGRD4RkYzWf1vfsDpTT4lIgIikisjB1nP5cZ9+fXe8Ry8iocaYyta/PwokGWNWWxyr20TkOuDvrZPD/QLAGPMdi2P1iIiMB2zAc8DjxhiXmcyodUnME7RZEhNY5oglMfuDiMwHqmlZFW6i1Xl6qnUVu+HGmM9FJARIB25xxf8vretuBxljqkXEF9gJfMMYs6cvvr5bXtFfLPlWQXSy2ImzM8ZsNcZcXOl8Dy3z+bskY0yGMea41Tl66B/LaRpjGoCLS2K6JGPMp0C51Tl6yxhz2hjzeevfq4AMOlnBztmZFtWtL31b//RZb7ll0QOIyFMikg/cA/zQ6jx94MvA+1aH8FB2L4mprCEi8cA0YK/FUXpMRLxF5ABQAnxojOmzc3HZoheRj0TkSAd/lgAYY75njIkBXgcesTZt57o6j9Z9vgc00XIuTsuec3FRdi+JqfqfiAQDbwOPtftt3qUYY5qNMVNp+c09RUT67LaaPUsJOiVjzDV27vonYDPwpAPj9FhX5yEi9wM3AVcbJ/9ApRv/T1yNPctpKgu03s9+G3jdGPMXq/P0BWNMhYhsAxYCffKBucte0V+KiIxp83IxcMyqLL0hIguB7wCLjTG1VufxYLokphNq/QDzBSDDGPNbq/P0hohEXBxVJyIDgGvow95y11E3bwPjaBnlkQesNsYUWpuq+0QkE/AHylrf2uOKo4cARORW4H+BCKACOGCMud7SUN0gIouA3/HPJTGfsjZRz4nIG8CVtEyJewZ40hjzgqWhekBELgN2AIdp+VkH+K4x5j3rUvWMiEwGXqHl35cXsNEY81999vXdseiVUkr9k1veulFKKfVPWvRKKeXmtOiVUsrNadErpZSb06JXSik3p0WvlFJuToteKaXc3P8H5U1S4GJeGygAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(-3,3,100)\n",
    "y_norm = stats.norm.pdf(x)\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "ax.plot(x,y_norm);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Parametri del set di dati"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "DescribeResult(nobs=100, minmax=(0.0044318484119380075, 0.3987591533537418), mean=0.1645975096425618, variance=0.01964894623903066, skewness=0.4042398613429197, kurtosis=-1.3613112580554971)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats.describe(y_norm)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Grafico della distribuzione cumulativa:  $\\,\\,C(x) = \\int_{-\\infty}^x N(y)\\, dy$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "y_norm_cumulative = stats.norm.cdf(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "ax.plot(x,y_norm_cumulative);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 9.4.1 Distribuzione di Poisson: $f(k) = \\exp(-\\mu)\\, \\frac{\\mu^k}{k!}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Grafico della distribuzione"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "mu = 1.5\n",
    "k = np.arange(0,10,1)\n",
    "n = stats.poisson.pmf(k, mu)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "ax.scatter(k,n);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Parametri del set di dati"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "DescribeResult(nobs=10, minmax=(2.3638318270467896e-05, 0.33469524022264474), mean=0.09999959024990236, variance=0.015889019294170213, skewness=0.7954958214401758, kurtosis=-0.9316199632660513)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats.describe(n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Informazioni addizionali e buoni esempi: https://realpython.com/numpy-random-number-generator/"
   ]
  }
 ],
 "metadata": {
  "hide_input": false,
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
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  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.12"
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  "toc": {
   "base_numbering": 1,
   "nav_menu": {},
   "number_sections": true,
   "sideBar": true,
   "skip_h1_title": true,
   "title_cell": "Table of Contents",
   "title_sidebar": "Contents",
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   "toc_position": {},
   "toc_section_display": true,
   "toc_window_display": false
  },
  "varInspector": {
   "cols": {
    "lenName": 16,
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   },
   "kernels_config": {
    "python": {
     "delete_cmd_postfix": "",
     "delete_cmd_prefix": "del ",
     "library": "var_list.py",
     "varRefreshCmd": "print(var_dic_list())"
    },
    "r": {
     "delete_cmd_postfix": ") ",
     "delete_cmd_prefix": "rm(",
     "library": "var_list.r",
     "varRefreshCmd": "cat(var_dic_list()) "
    }
   },
   "types_to_exclude": [
    "module",
    "function",
    "builtin_function_or_method",
    "instance",
    "_Feature"
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   "window_display": false
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 },
 "nbformat": 4,
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}