diff --git a/_toc.yml b/_toc.yml index 01021af93e0d986f0cbb208028356328fc73a810..5e81b5a646c044bf82078bacb58cadbded02fc68 100644 --- a/_toc.yml +++ b/_toc.yml @@ -14,3 +14,9 @@ parts: - file: zufallsvariablen-verteilungen - file: erwartungswert - file: numpy +- caption: Stetige Stochastik und Scipy + numbered: true + chapters: + - file: stetige-verteilungen + - file: normalverteilung + - file: wichtigste-stetige-verteilungen diff --git a/images/ProbOnto2.5.jpg b/images/ProbOnto2.5.jpg new file mode 100644 index 0000000000000000000000000000000000000000..299b5bd642a24365dd73603677753ee977991568 Binary files /dev/null and b/images/ProbOnto2.5.jpg differ diff --git a/normalverteilung.ipynb b/normalverteilung.ipynb new file mode 100644 index 0000000000000000000000000000000000000000..a93876643ca1a0d1d32de0646ad5358039a7e8c3 --- /dev/null +++ b/normalverteilung.ipynb @@ -0,0 +1,337 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "3fdf0f52-1f02-4b42-8152-4e632412fd8c", + "metadata": {}, + "source": [ + "# Die Gausssche Normalverteilung" + ] + }, + { + "cell_type": "markdown", + "id": "3ddc6bdb-9038-41d4-b999-5e80d347b298", + "metadata": {}, + "source": [ + "## Approximation der Binomialverteilung" + ] + }, + { + "cell_type": "markdown", + "id": "af111971-55f2-4e9e-861c-4874ece317c1", + "metadata": {}, + "source": [ + "Betrachten wir eine Folge von Bernoulli-verteilten Zufallsvariablen $X_i$ mit gleichem Parameter $p$ und $S_n := \\sum_{i=1}^n X_i$ die Summe über die ersten $n$ davon, so wissen wir bereits, dass $S_n \\sim Bin(n,p)$, also dass diese Summe Binomialverteilt ist.\n", + "\n", + "Für die konkreten Werte $n=100,\\ p=0.4$ sieht ein Histogramm von $100000$ Samples etwa so aus:" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "id": "0ae9357f-96be-4180-8fbb-e5b4269abad5", + "metadata": { + "slideshow": { + "slide_type": "skip" + } + }, + "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": [ + "import numpy as np\n", + "from scipy.stats import binom, norm\n", + "from matplotlib import pyplot as plt\n", + "np.random.seed(123123)\n", + "binom_samples = binom.rvs(n=100,\n", + " p=0.4,\n", + " size=100000)\n", + "plt.hist(binom_samples, bins=40, density=True)\n", + "plt.xlabel(\"$x$ binomialverteilt\")\n", + "plt.ylabel(\"Dichte\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "cf7b3ed2-f44a-4250-9cc4-ee10130f22cb", + "metadata": {}, + "source": [ + "Da diese Form für große Stichproben stets so aussieht, unabhängig von $p$ (probieren Sie das aus!), könnte man sich vorstellen, dass die Verteilung sich approximieren lässt mit Hilfe einer Funktion, die leichter zu berechnen ist als Binomialkoeffizienten.\n", + "\n", + "Tatsächlich gibt es zur Approximation von Binomialkoeffizienten auch die *Stirling-Formel*\n", + "\n", + "$$\n", + "n! \\approx \\sqrt{2\\pi n} {\\left(\\frac{n}{e}\\right)}^n\n", + "$$\n", + "\n", + "Damit lässt sich nun (auch wenn es nicht ganz einfach ist) eine Approximation zeigen:" + ] + }, + { + "cell_type": "markdown", + "id": "3000f027-bf66-4548-8221-0b5af436c558", + "metadata": {}, + "source": [ + "$$\n", + "P(S_n = k) \\sim\n", + "\\frac{1}{\\sqrt{np(1-p)}} \\frac{1}{\\sqrt{2\\pi}} e^{-\\frac{x^2}{2}} \\quad \\text{ fuer } n \\to \\infty\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "7a0e7875-c32a-49d2-aa2c-4ffcde8e40f2", + "metadata": {}, + "source": [ + "wobei für $x$ auf der rechten Seite die Folge $x=x_n$ mit\n", + "$x_n = x_n(k) = \\frac{k-np}{\\sqrt{np(1-p)}}$ eingesetzt werden muss. Das Symbol $\\sim$ bedeutet: asymptotisch gleich, d.h. der Quotient konvergiert gegen $1$.\n", + "\n", + "Diese Approximation ist insofern hilfreich, als dass wir nun die Funktion $e^{-\\frac{x^2}{2}}$ tabellieren können und konkrete Werte für gewisse $n,p$ ablesen können. Das ist wesentlich effizienter, als Binomialkoeffizienten auszurechnen. Die ganze Beobachtung heißt auch **Satz** von deMoivre-Laplace und soll uns zunächst als Motivation dienen, die Funktion $e^{-\\frac{x^2}{2}}$ näher zu untersuchen." + ] + }, + { + "cell_type": "markdown", + "id": "340c8b43-2f67-4d19-877e-912d783f5994", + "metadata": {}, + "source": [ + "## Ein Integral\n", + "\n", + "Um aus der Funktion $e^{-\\frac{x^2}{2}}$ eine Wahrscheinlichkeitsdichte zu machen, muss das Integral $1$ ergeben. Wir berechnen, weil es danach sehr nützlich wird, gleich ein etwas allgemeineres Integral:\n", + "\n", + ":::{admonition} Lemma\n", + "Sei $v \\in \\mathbb{R}$ und $v > 0$. Dann gilt\n", + "\n", + "$$\n", + "\\int_{-\\infty}^\\infty e^{-\\frac{x^2}{2v}} dx = \\sqrt{2\\pi v}.\n", + "$$\n", + ":::\n", + "\n", + ":::{admonition} Beweis\n", + "Da auf der rechten Seite eine Quadratwurzel steht, bietet es sich an, beide Seiten der Gleichung zu quadrieren. Wir formen die linke Seite dann weiter um, bis wir ein Integral über $\\mathbb{R}^2$ in Polarkoordinaten transformieren können und dann leicht Stammfunktionen bestimmen können:\n", + "\n", + "$$\n", + " & {\\left( \\int_{-\\infty}^\\infty e^{-\\frac{x^2}{2v}} dx \\right)}^2 \\\\\n", + " =& {\\left( \\int_{-\\infty}^\\infty e^{-\\frac{x^2}{2v}} dx \\right)}{\\left( \\int_{-\\infty}^\\infty e^{-\\frac{y^2}{2v}} dy \\right)} \\\\\n", + " =& \\int_{-\\infty}^\\infty e^{-\\frac{x^2}{2v}} {\\left( \\int_{-\\infty}^\\infty e^{-\\frac{y^2}{2v}} dy \\right)} dx \\\\\n", + " =& \\int_{-\\infty}^\\infty {\\left( \\int_{-\\infty}^\\infty e^{-\\frac{x^2}{2v}} e^{-\\frac{y^2}{2v}} dy \\right)} dx \\\\\n", + " =& \\int_{-\\infty}^\\infty {\\left( \\int_{-\\infty}^\\infty e^{-\\frac{x^2+y^2}{2v}} dy \\right)} dx \\\\\n", + " =& \\int_{\\mathbb{R}^2} e^{-\\frac{x^2+y^2}{2v}} d(x,y) \\\\\n", + " =& \\int_{0}^\\infty \\int_{0}^{2\\pi} r e^{-\\frac{r^2}{2v}} d\\phi dr \\\\\n", + " =& \\int_{0}^\\infty {\\left[ r e^{-\\frac{r^2}{2v}} \\phi \\right]}_{\\phi=0}^{\\phi=2\\pi} dr \\\\\n", + " =& \\int_{0}^\\infty r e^{-\\frac{r^2}{2v}} (-2\\pi) dr \\\\\n", + " =& -2\\pi {\\left[ -v e^{-\\frac{r^2}{2v}} dr \\right]}_{r=0}^{r=\\infty} \\\\\n", + " =& 2\\pi v\n", + "$$\n", + ":::" + ] + }, + { + "cell_type": "markdown", + "id": "05a03b80-9d5c-4ee8-94d4-38d12fc7d48f", + "metadata": {}, + "source": [ + "Da Integrale in der Statistik häufiger vorkommen, ist es nicht ganz verkehrt, sich bei dieser Rechnung klar zu machen, was genau warum in jedem Schritt passiert." + ] + }, + { + "cell_type": "markdown", + "id": "9fc75e9c-a648-476a-a297-97aa9e313f71", + "metadata": {}, + "source": [ + "## Die eindimensionale Normalverteilung\n", + "\n", + ":::{admonition} Definition\n", + "Seien $\\mu,\\sigma \\in \\mathbb{R}$ mit $\\sigma > 0$. Dann heißt die Verteilung mit Dichtefunktion\n", + "\n", + "$$\n", + "\\phi(x) = \\phi(x,\\mu,\\sigma) = \\frac{1}{\\sigma\\sqrt{2\\pi}} e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}}\n", + "$$\n", + "\n", + "*Normalverteilung* und wir schreiben für eine reelle Zufallsvariable $X$ mit dieser Verteilung auch $X \\sim \\mathcal{N}(\\mu,\\sigma^2)$.\n", + ":::" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "3465b513-ac20-44a5-8c02-906058b2e6af", + "metadata": { + "tags": [] + }, + "outputs": [ + { + "data": { + "image/png": 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xIjJcRL4XkZUi8rifcRpjjPGPb4nGG1/+VaAnbqjPASLSIqzYncBuVW0CDAVe8KZfBySo6tlAO+DurCRkjDGmePGzRtMeSFXVdd6Y1iOAPmFl+gDDveejgW4iIoACFUQkDjcudzpu/HljjDHFjJ/XaOoCm0JebwY6ZFdGVTNEZC9uhMfRuCS0FSgPPKiqu8IXICIDgYEADRo0KOz4jSkcqnRdm0LP1bM57cBOah3YRZnjGcxs3I7Pz+wMmT0hxvrlmJIrWrfu9sBx4HSgEfCwiDQOL6Sqw1Q1WVWTExN9HU7BmIKZPRsuuoh3Rz9F9zVzqXp4P5ur1GZj1dO4edFkRn/4GDRsCFOnBh2pMb7xs0azBagf8rqeNy1Smc1eM1kVYCdwIzBFVY8BP4vIbCAZN5a3MdHv+HH43e/gtdfgtNP402X3MvKc7hyLjT9RpOLRQ1yydj7/Wv859OgBTzwBTz8NcdYZ1JQsftZo5gNNRaSRiJQB+gPjw8qMB271nvcDZqiqAj8ClwCISAWgI7DKx1iNKTzHjsEtt7gk8+CDkJrKB217nZRkAA4klGd8i4to3v0pPm7dHZ57jnmN2nDOH0aQNHjSiYcxxZ1viUZVM4BBwFRgJTBKVZeLyBAR6e0VewuoISKpwENAVhfoV4GKIrIcl7DeUdWlfsVqTKE5cgSuvRY+/hj+9jd46SWoUCHnt8SX5fGev+cPVz5Mm62r+c/YZymTcayIAjbGf+IqEMVfcnKy2lDOJihJgyeBKq+N+xs9f/iWP112Lx+0zf8PMXuvmMm/JrzIpy0v5qFeD4GI/aDT+EpEFqhqsp/LsMZgYwrJbQsm0POHb3m26x0FSjIA41t0pf6e7fzx6/fZVOU0hl5wUyFHaUzRs0RjTCE4e+sanvjybaY3ac8b7a8+pc96tdP1NNizjT98+zErajcCrEZjijdLNMbkUXYX5isdPcjE8S+QVqEaj1zxIIic2oJE+J/L76flz+t4ZtprsPsRqFbt1D7TmABF6+9ojCk2np7+OnX3/szvej/K3nKVCuUzM2LjeKzn76l+aC88/HChfKYxQbFEY8wpOG/TMq5Z/iX/7ngdC+udVaifvbz2Gfynw7XwzjswfXqhfrYxRckSjTEFFJN5nKe+GMaWSon8u9N1vizjX+cPgDPPhIED4cABX5ZhjN8s0RhTQP2XTqPlz+t47uI7OBJf1pdlHI0rA2+9BRs2wAsv5FremGhkicaYAqh85ACPfPU+c+u3YlLzLv4u7Pzz4YYbYOhQ2L7d32UZ4wNLNMYUwAPffESVIwd4+tKBp97LLC+eecbddeC55/xfljGFzLo3G5NPtffv4KbFkxnZ+jJW1vrVTcULXVa36udaXcq1r/6bS9LbsKVKrRPz7c4BJtpZjcaYfLp73qfEqPLvTtcX6XL/1XkAIDzwzUdFulxjTpUlGmPyoebB3dy4ZApjW17M5iq1i3TZ2yrXZHjbK7lm+Qya7PixSJdtzKmwRGNMPvz2u7HEH8/g3x396c6cm9c69uNwfAL3zv0kkOUbUxCWaIzJo2qH9nLLoslMOOsCNlSvG0gMu8tXYdTZl9F75Vectm9HIDEYk1+WaIzJoztSxlPh2BFe7Vi012bCvX1eH2JUuW3hhEDjMCavLNEYkxcHD/KbhRP5vFln1iQ2DDSUzVVq8/mZ53Pj4ilUOHoo0FiMyQtfE42I9BCR1SKSKiKDI8xPEJGR3vx5IpLkTb9JRBaHPDJFpI2fsRqTow8/pMrRg7x1Xp+gIwFgWPurqXz0IDcstXugmejnW6IRkVjckMw9gRbAABFpEVbsTmC3qjYBhgIvAKjqh6raRlXbALcA61V1sV+xGpMjVXj5ZZbXakxK3fBNOBhL6zRjXr2W3JHyGWRkBB2OMTnys0bTHkhV1XWqmg6MAMJPB/sAw73no4FuIr/6mfUA773GBOOrr2DZMoa3vbJo7gKQR2+0v4Z6+36GMWOCDsWYHPmZaOoCm0Jeb/amRSyjqhnAXqBGWJkbgI99itGY3L38MlSvzmctLgo6kpP8t8l5bKhaB159NehQjMlRVHcGEJEOwCFVXZbN/IEikiIiKWlpaUUcnSkVNm2CcePgt7/laHxC0NGcRCWGj9tcDl9/DStXBh2OMdnyM9FsAeqHvK7nTYtYRkTigCrAzpD5/cmhNqOqw1Q1WVWTExMTCyVoY07y+uvuGs299wYdSUSjW10K8fEwbFjQoRiTLT8TzXygqYg0EpEyuKQxPqzMeOBW73k/YIaqKoCIxADXY9dnTFDS0+GNN+CqqyApKehoItpZoSpcfTUMH+7u7mxMFPIt0XjXXAYBU4GVwChVXS4iQ0Skt1fsLaCGiKQCDwGhXaAvBDap6jq/YjQmRxMnQloa3H130JHkbOBA2L0bRo8OOhJjIhKvAlHsJScna0pKStBhmJLkyith8WLYuBFiY0/crj/abHiupxvuuU4d10POmHwQkQWqmuznMqK6M4AxgfnpJ/j8c7j1VoiNDTqanMXEwF13uU4BK1YEHY0xv2IDnxnjCa2x3Dv3Ex7LzOSiHUlsjNKazEluuw3+9Cd3TWno0KCjMeYkVqMxJpwq/b7/gnn1W7Gx2ulBR5M3tWq5TgsffQTHjgUdjTEnsRqNMWHabVnJGbu28O+A79KcV1k1sctowRs/f8rtNz7Ll2ecB9gwzyY6WI3GmDDXL53OgTLlmHzm+UGHki8zG7djV7nKXLNsRtChGHMSSzTGhCiXfoQrV33NpDO7cLhM2aDDyZdjsfFMOOsCuq+ZS+UjB4IOx5gTLNEYE+Ky1HlUOHaEMWd3CzqUAvm05SUkHD9Gz9Wzgw7FmBMs0RgTos+KmWyplMj8etExHEB+LanTjLXV63LN8i+DDsWYEyzRGOOpdmgvF65fyIQWF6JSTHcNET5teQkdNi2j3p5tQUdjDGCJxpgTeq2eTXzm8agbDiC/xrW8GICrrVZjooQlGmM8vVfMZHXNBqxMbBR0KKdkS5VazKvfij4rZrk7TxsTMEs0xgBs3Ej7zSv4rEXXqBpFs6AmnHUhTXZthqVLgw7FGEs0xgDwsRv2aPxZFwYcSOH4vFlnMiQGRo4MOhRjLNEYA8CHH5JS9yw2Vz0t6EgKxc4KVfm24TkwYoQ1n5nAWaIxZvlyWLas2HcCCDfhrAtg/Xqw4TNMwCzRGDNqFMTE8Hkxu+VMbqY26+yGeR5hg9SaYFmiMaWbKnzyCVx4ITsqVAs6mkK1r2xFuPxyl0gzM4MOx5RiviYaEekhIqtFJFVEBkeYnyAiI73580QkKWReaxGZIyLLReR7ESleN54yxcPy5bByJVxfPO7UnG/9+8PmzfDtt0FHYkox3xKNiMQCrwI9gRbAABEJv6/HncBuVW0CDAVe8N4bB3wA3KOqLYGugA2yYQrfJ5+4ESqvuSboSPzRuzeULWvNZyZQftZo2gOpqrpOVdOBEUCfsDJ9gOHe89FANxERoDuwVFWXAKjqTlU97mOspjRSdc1KF14ItWsHHY0/KlWCK66AMWOs+cwExs9EUxfYFPJ6szctYhlVzQD2AjWAZoCKyFQRWSgij0ZagIgMFJEUEUlJS0sr9C9gSrjly2HVqpLbbJalXz/Yts2az0xgorUzQBzQBbjJ+3u1iPzqvu2qOkxVk1U1OTExsahjNMWd19usxDabZenVC8qUcbUaYwLgZ6LZAtQPeV3PmxaxjHddpgqwE1f7+UpVd6jqIWAy0NbHWE1pE9LbrMQ2m2WpXNn1Phszxn68aQIR5+NnzweaikgjXELpD9wYVmY8cCswB+gHzFBVFZGpwKMiUh5IBy7CdRYwpsCSBk868bxZ2gamrVrFn+pfzAch00usfv1gwgSYPx/atw86GlPK+Faj8a65DAKmAiuBUaq6XESGiEhvr9hbQA0RSQUeAgZ7790NvIRLVouBhapaCo4Gpqj0+GEOmQhTm3UKOpSicdVVEBdnzWcmEH7WaFDVybhmr9BpT4Y8PwJcl817P8B1cTam0PX44VsW1D2LtIrVgw6laFSrBpdeCqNHw/PPl4g7VJviI1o7Axjjmwa7t9Li5/VMKS21mSzXXgvr1sGSJUFHYkoZX2s0xkSjy3+YA1Aqms1Cr0tVO1Se+RLDa/c9z/9deAsAG57vFVRophSxGo0pdXr88C3Lap9RYoYEyKvd5aswr0Ereq6eHXQoppSxRGNKlVr7d9Lup1V83qxz0KEEYkqzzjTZtZkzdm7KvbAxhcQSjSlVuq+ZC7gDbmk0vUlH4JfmQ2OKgiUaU6r0+OFbUqvXY23N+rkXLoG2Va7J4jpN6b7GEo0pOpZoTKlR9fA+Ov74PVPOLJ21mSzTmnaizdY1nLZvR9ChmFLCEo0pNbqlzidOM5natOT3NstJVm87q9WYomKJxpQal6+Zw0+VavL9aU2CDiVQa2vUJ7V6PS63RGOKiCUaUzocOsQF6xcxrWlH+1U8rlbT4cdlsHNn0KGYUsASjSkdpk2jXMZRl2gMU5t1Ik4zYeLEoEMxpYAlGlM6jB3LnrIV+a5+q6AjiQpLT2vKT5VqwtixQYdiSgFLNKbky8iACRP47xnnkRFrd10CQMTV7qZOhYMHg47GlHCWaEzJ9/XXsHs300p5b7Nw05p2hCNHYPr0oEMxJZwlGlPyjR0LZcvyVSMbpDXUd/VbQdWqMG5c0KGYEs4SjSnZVN2BtHt3DpcpG3Q0USUjNg569XIdAjIygg7HlGC+JhoR6SEiq0UkVUQGR5ifICIjvfnzRCTJm54kIodFZLH3eN3POE0JtmgRbNoEffsGHUl06tvXdXGebXd0Nv7xLdGISCzwKtATaAEMEJEWYcXuBHarahNgKPBCyLy1qtrGe9zjV5ymhBs3DmJi4Morg44kOl1+OSQkWPOZ8ZWfNZr2QKqqrlPVdGAE0CesTB9guPd8NNBNxH5NZwrRuHFw/vmQmBh0JNGpUiU3xPO4ca6Z0Rgf+Jlo6gKhg15s9qZFLKOqGcBeoIY3r5GILBKRWSJyQaQFiMhAEUkRkZS0tLTCjd4Uf+vWwfffW7NZbvr2hQ0b3LoyxgfR2hlgK9BAVc8FHgI+EpHK4YVUdZiqJqtqcqKdsZpwn33m/vYJr0ibk1x1lbstjzWfGZ/4+eu1LUDooB/1vGmRymwWkTigCrBTVRU4CqCqC0RkLdAMSPExXlMCJA2edOL5iI/epmrNhvR4YxWwKrigol3t2tCpk+sG/uSTQUdjSiA/E818oKmINMIllP7AjWFlxgO3AnOAfsAMVVURSQR2qepxEWkMNAXW+RirKWGqHdrLeZtX8GrH64IOJaplJeaBZc7kiW/f4fx732FLlVoAbHi+V5ChmRLEt6Yz75rLIGAqsBIYparLRWSIiPT2ir0F1BCRVFwTWVYX6AuBpSKyGNdJ4B5V3eVXrKbkuWRtCrGayXS7iWaeZK2nS1PnBRyJKYnyVKMRkWbAa0BtVW0lIq2B3qr615zep6qTgclh054MeX4E+NUpp6qOAcbkJTZjIum+Zg5bK9Yo9WPP5NX66nVZU6M+3dfMYXi7q4IOx5Qwea3RvAE8DhwDUNWluKYwY6JO2WNHuHD9IneWbr3l82xa0450+HEZlY8cCDoUU8LkNdGUV9XvwqbZPStMVOqyYYmNPVMA05t2JE4zuWTt/KBDMSVMXhPNDhE5A1AAEemH64JsTNTpvmYO+xIqMK+BjT2TH0vqNGV7xep0/8GGeDaFK6+9zu4HhgHNRWQLsB64ybeojCmgmMzjdEv9jhlnJHMsNj7ocIoVlRi+aNKevstnkpCRHnQ4pgTJa41GVfVSIBForqpd8vFeY4pMuy0rqXF4H9ObWLNZQUxr2okKx47QeeOSoEMxJUhek8UYAFU9qKr7vWmj/QnJmILrvmYuR2PjmNW4XdChFEtzGrRmf5ly1nxmClWOTWci0hxoCVQRkWtCZlUGbHAPE11U6b5mLt82PIcDCeWDjqZYSo+LZ1bjZC5N/Q6OH4fY2KBDMiVAbjWaM4ErgarAVSGPtsBdvkZmTH4tW0bDPdtsyOZTNK1pRxIP7YG5c4MOxZQQOdZoVPUz4DMR6aSqVpc20W3cODIRvmjSIehIirUvz0gmPSaOMllDLBhzivJ6jSZVRJ4QkWEi8nbWw9fIjMmvceNYdPqZpFWsFnQkxdr+hArMadja3WTTxqgxhSCvieYz3J2VvwAmhTyMiQ4//ggLFzKtmfU2KwxTm3WCtWth+fKgQzElQH7uDPCYqo5S1TFZD18jMyY/vLFnrFtz4ZjepIO7fc/YsUGHYkqAvCaaiSJyha+RGHMqxo6Fs85iXY16QUdSIqRVrA4dO9pgaKZQ5JhoRGS/iOwD/oBLNodFZF/IdGOCt2MHzJoF11yTe1mTd337wsKFsHFj0JGYYi7HRKOqlVS1svc3RlXLhbz+1dDKxgRiwgTIzISrrw46kpIla31mDYltTAHlqelMRK4WkSohr6uKSF/fojImP8aOhQYNoG3boCMpWZo2hZYt7TqNOWV5vUbzF1Xdm/VCVfcAf8ntTSLSQ0RWi0iqiAyOMD9BREZ68+eJSFLY/AYickBEHsljnKa0OXAApk1zzTw29kzh69sXvvrKNU8aU0B5TTSRyuV2+5pY4FWgJ9ACGCAiLcKK3QnsVtUmwFDghbD5LwGf5zFGUxpNmQJHj9r1Gb9cc41rlpwwIehITDGW10STIiIvicgZ3uMlYEEu72kPpKrqOlVNB0YAfcLK9AGGe89HA91E3Gmp1zS3HrCO/CZ7n34KNWtCly5BR1IynXsuNGwIY+zXDKbg8ppofgekAyO9x1HcGDU5qQtsCnm92ZsWsYyqZgB7gRoiUhF4DHg6j/GZ0ig9HSZNgt697eaPfhFxtZrp02GfdTQ1BZOnROMNDzBYVZO9x+OqetDHuJ4ChqpqjoOXi8hAEUkRkZS0tDQfwzFRacYMd/CzZjN/XXutS+qTJwcdiSmmcrvO8g9VfUBEJuAN4xxKVXvn8PYtQP2Q1/W8aZHKbBaRONxtbnYCHYB+IvK/uDtHZ4rIEVV9JWz5w3Ajf5KcnGw3ZSptPv0UKlaEbt2CjqRk69QJTjvNNZ/17x90NKYYym0o5/e9vy8W4LPnA01FpBEuofQHbgwrMx64FZgD9ANmqKoCF2QVEJGngAPhScaUchkZ7lfrV14JZW1oJF/FxLjeZ++9B4cPQ7lyQUdkipnchglY4P2dJSKJ3vM8tVGpaoaIDAKmArHA26q6XESGACmqOh54C3hfRFKBXbhkZEy2kga7e7l22riUj9PSuOdII6YMtvu7+iEpZL122VWXDw4d4q6bn2N6U3c/uQ3P9woqNFPM5FajyapRDMJdzxERyQBeVtUhub1XVScDk8OmPRny/AhwXS6f8VRuyzGlT8/VszkUn8BMG7K5SMxtcDZ7ylakx+rZJxKNMXmV273OHgLOB85T1eqqWg13/eR8EXmwKAI0JpxoJj1++JaZjdpxJN6azYpCRmwcXzTpwKWp3xF//FjQ4ZhiJrdeZ7cAA1R1fdYEVV0H3Az8xs/AjMlOuy0rqXVwN5+faaM/FqUpzTpT5ehBOm9cGnQoppjJLdHEq+qv7j3hXaeJ9yckY3J2xarZHI2NZ8YZ5wUdSqnydaNz2V+mHFes+iboUEwxk1uiSS/gPGN8kdVs9lWjthxMKB90OKXK0bgyfNGkA93XzCXueEbQ4ZhiJLdEc443/kz4Yz9wdlEEaEyoc7au4fT9O/j8zM5Bh1IqTWp+AdWO7KfzxiVBh2KKkdy6N9t9PUxU6bl6Nukx7sK0KXpZzWe9rPnM5ENe73VmTPBU6bXqa75JasO+shWDjqZUCm0+45j1PjN5Y4nGFB9z51JvXxoTz7og97LGN5Obd6Hakf3uXnPG5IElGlN8jBzJ0dg4+8FgwL5q1Jb9ZcrBqFFBh2KKCUs0pnjIzIRPPmFW42T2J1QIOppSLav5jHHjrPnM5IklGlM8zJ4NP/3ExObWbBYNJjfvArt2wX//G3QophiwRGOKh5EjoWxZvmjSPuhIDDCrUTuoUgU+/jjoUEwxYInGRL/jx2H0aOjVi0Nl7Bb10SA9Lt4NODd2rBs6wJgcWKIx0W/WLNi+HW64IehITKgBA2D/fht50+TKEo2JfiNHQoUK0MvGP4kqF18MtWpZ85nJlSUaE93S0+GTT6BPHyhv9zaLKnFxcP31MHEi7NsXdDQmilmiMdHt889h92646aagIzGRDBgAR4+6rs7GZMPXRCMiPURktYikisjgCPMTRGSkN3+eiCR509uLyGLvsURErvYzThPFPvgAEhPhssuCjsRE0qkTNGwIH30UdCQmivmWaEQkFngV6Am0AAaISIuwYncCu1W1CTAUeMGbvgxIVtU2QA/gPyKS67DTpoTZuxcmTHCdAOJt+KOoJOJqNV98AWlpQUdjopSfNZr2QKqqrlPVdGAE0CesTB9guPd8NNBNRERVD6lq1oAXZQH1MU4TrT791DXLWLNZdBswwHVBHzky6EhMlPKzllAX2BTyejMQfm/3E2VUNUNE9gI1gB0i0gF4G2gI3BKSeE4QkYHAQIAGDRoU+hcwRS9p8KQTzz8c8Q/qVq1D17E7YNykHN5lAtW6tXu8/z4MGhR0NCYKRW1nAFWdp6otgfOAx0WkbIQyw1Q1WVWTExMTiz5I45va+3fQaeP3fNaiq2ueMdHtN7+B776DVauCjsREIT8TzRagfsjret60iGW8azBVgJ2hBVR1JXAAaOVbpCbqXLXyK2JQxrXsGnQoJi9uvBFiYlytxpgwfjadzQeaikgjXELpD9wYVmY8cCswB+gHzFBV9d6zyWtOawg0Bzb4GKuJMtcs/5IlpzVlffW6QYdishHazAnwbsNzafryG3TJ6IhKDBuetx/YGse3Go13TWUQMBVYCYxS1eUiMkREenvF3gJqiEgq8BCQ1QW6C7BERBYDY4H7VHWHX7Ga6NJy+1pa/Lye0Wd3CzoUkw+ftrqEuvvT6PjjsqBDMVHG1y7DqjoZmBw27cmQ50eA6yK8733A6uClVL/vv+BobBzjz7oo6FBMPkxt2pF9ZcpzzbIZzGnYOuhwTBSJ2s4ApnQqk3GMvstnMr1pJ/aWqxR0OCYfjsYnMLl5F3r+MJty6UeCDsdEEUs0Jqp0S51HtSP7+eTsS4MOxRTAp60uoWL6YS5fMyfoUEwUsURjosp133/B1oo1+DqpTdChmAKYX68FG6rW4Yal04IOxUQRSzQmemzdykXrF/Jpq0vIjIkNOhpTACoxjGp9GZ1+/B7WrAk6HBMlLNGY6PH++8RqJqOt2axY++TsS8mQGHjrraBDMVHCEo2JDqrw9tvMr9vCfjtTzKVVrM6MJu3h3Xfh2LGgwzFRwBKNiQ6zZsHq1Xzc5vKgIzGFYETr7m747YkTgw7FRAFLNCY6vP46VKvGpDO7BB2JKQSzGreD00+HN94IOhQTBSzRmOBt3+6GBLjtNo7GJwQdjSkEx2Ni4Y47YMoU2LQp9zeYEs0SjQneO++4tvyBA4OOxBSmO+5wf998M9g4TOAs0ZhgZWbCf/4DXbtC8+ZBR2MKU6NG0KMHDBsG6elBR2MCZInGBGvaNNiwAe65J+hIjB8GDYJt21zTqCm1LNGYYL3+OiQmwtVXBx2J8UOPHtC4MbzyStCRmABZojHBWb8eJkyAO++EMmWCjsb4ISYG7r8fZs+GRYuCjsYExNdhAozJ0Suv/HIgMiVO1sBolY/UZW58AuNvG8zgnr8HsEHRShmr0Zhg7N/veiNddx3Uqxd0NMZH+8pWZFyLi+m7YiZVDu8POhwTAF8TjYj0EJHVIpIqIoMjzE8QkZHe/HkikuRNv0xEFojI997fS/yM0xStpMGTeOrqR2DfPvrEJZM0eNKJhymZ3mvbi7IZ6fRfOjXoUEwAfEs0IhILvAr0BFoAA0SkRVixO4HdqtoEGAq84E3fAVylqmcDt2KjbZYoMZnHuW3BBBac3pwlp58ZdDimCKyq1Yg5Dc7mtpQJxB+3+5+VNn7WaNoDqaq6TlXTgRFAn7AyfYDh3vPRQDcREVVdpKo/edOXA+VExH4yXkJcsjaFpD1beTs5fHMwJdl/2l9LnQM7uWrlV0GHYoqYn4mmLhB674nN3rSIZVQ1A9gL1Agrcy2wUFWPhi9ARAaKSIqIpKSlpRVa4MZfd6R8xpZKiUw5s3PQoZgiNLNxO1bVbMjAeZ+6u3WbUiOqOwOISEtcc9rdkear6jBVTVbV5MTExKINzhTM/Pl0/nEp77a7yt0Py5QeIgzrcA3Nd2x090AzpYafiWYLUD/kdT1vWsQyIhIHVAF2eq/rAWOB36jqWh/jNEXpuefYU7YiH7XpEXQkJgATzrqQrRVrwN//HnQopgj5mWjmA01FpJGIlAH6A+PDyozHXewH6AfMUFUVkarAJGCwqs72MUZTlJYtg3HjeLfdVRxMKB90NCYAx2Ljeeu8PvDll5CSEnQ4poj4lmi8ay6DgKnASmCUqi4XkSEi0tsr9hZQQ0RSgYeArC7Qg4AmwJMisth71PIrVlNEnn8eKlTg3XZXBR2JCdCIc3pAlSpuezClgmgJuSiXnJysKXaGFL3WroVmzeChh0iK7Rp0NCZgG8rMg2eegSVLoHXroMMp1URkgaom+7mMqO4MYEqQF16A+Hh46KGgIzHR4MEHoXJlGDIk6EhMEbBEY/z3448wfDjcfjvUqRN0NCYaVKsGDzwAY8a4Wo0p0SzRGP899RSIwBNPBB2JiSYPPOCu1Tz9dNCRGJ9ZojH+WrnS1Wbuuw/q18+9vCk9smo1Y8fC4sVBR2N8ZInG+OvPf4by5eHxx4OOxESjrFrNk08GHYnxkSUa45+UFNcG//DDbhRNY8JVrQqPPeYGwJs5M+hojE8s0Rj//M//QI0a1tPM5OyBB9yYRI88ApmZQUdjfGAjbBp/TJvmHi++6LqxGhMifOyhq8+5nqGTXuIPfR7ls5YX2wicJYwlGlPomvzxM6a8PYjYanW4fOsZpNuAZiYX41p25Y6Uz3h01ntMaWZ39S5prOnMFLpbF06kya7NPHPJXaTHxQcdjikGVGJ49pI7qbs/jTtTPgs6HFPILNGYwrV9O3/45iO+bNyOGWecF3Q0phiZ26A105p25P45o2DTptzfYIoNSzSmcD3xBGUz0nnmkrvcjzSNyYch3e4iRhV+//ugQzGFyBKNKTxz58I77/B2cm/W1agXdDSmGNpcpTb/PH8AjBsH48NHFTHFlSUaUziOHIE77oD69Xm5c/+gozHF2Jvn9YVWrWDQIDhwIOhwTCGwRGMKxzPPuNvNDBtmg5qZU5IRGwevv+6u0zz1VNDhmEJgicacukWL3DAAt90Gl18edDSmJDj/fBg4EIYOhW+/DToac4p8TTQi0kNEVotIqogMjjA/QURGevPniUiSN72GiHwpIgdE5BU/YzSn6Ngxd/v/xER46aWgozElyYsvQsOGcMstsH9/0NGYU+BbohGRWOBVoCfQAhggIi3Cit0J7FbVJsBQ4AVv+hHgz8AjfsVnCsmTT7rxRF57zd2N15jCUqkSvPcerF9vtzEq5vy8M0B7IFVV1wGIyAigD7AipEwf4Cnv+WjgFRERVT0IfCMiTXyMz5yqadPcuO933QV9+wYdjSlBQm9R82iHa7nvzTf57a46fNG0g92ephjys+msLhD6q6vN3rSIZVQ1A9gL1MjrAkRkoIikiEhKWlraKYZr8mXbNtek0bIl/OMfQUdjSrChXW5iea3GvPD5P6mzz/bz4qhY3+tMVYcBwwCSk5M14HBKj+PHmd3xctru2stVVz1F6pAvg47IlGDHYuP5Xe9H+ey9B3lt3HPwUn8oWzbosEw++Fmj2QKEDqlYz5sWsYyIxAFVgJ0+xmQKw5//zPkbl/KXS+8mtWaDoKMxpcC6GvV4pNeDtNm6Bn73u6DDMfnkZ6KZDzQVkUYiUgboD4T/1Hc8cKv3vB8wQ1WtZhLN3nsP/vY3Pjrncka1vizoaEwpMrVZZ17pdD28+Sa88UbQ4Zh88K3pTFUzRGQQMBWIBd5W1eUiMgRIUdXxwFvA+yKSCuzCJSMARGQDUBkoIyJ9ge6qugITnG++cRf+L76YJ9vda/cyM0XupS43MajSHrj/fmjcGLp1CzokkwdSUioQycnJmpKSEnQYJde6ddChg+vCPHcuSf87J+iITCm1YfD5cMEFsHEjzJoF554bdEjFmogsUNVkP5dhdwYwudu0yZ05Hj8OEydC9epBR2RKs6pV4fPP3d+ePd1JkIlqlmhMzrZudUlm1y73u5lmzYKOyBioVw+mTnV3pujeHbaE9zMy0aRYd282PktLg0svhZ9+ckkm2dfatTF5EvpjznOveIL3Rv2ZXa2Suan/s2yuUtt+0BmFrEZjItuwwbWDr1vnmss62zjuJvosqtucm/o/S5UjBxj14WMk7bKaTTSyRGN+bckS6NQJtm+H6dOha9egIzImW0vrNGPAgOdIyEhn1EeDwToFRR3rdWZONn069OvHT1qGW697mjWJDYOOyJg8OWPHJoZ/8iT1jh2Ad9+F668POqRiwXqdmaKTmQnPPuvGk6lfn2tv/rslGVOsrK1Znz6/GQpt28INN8Bf/uK2axM4SzTG9Sjr3Rv+9CcYMADmzmVr5cSgozIm33ZWqAr//a8bI2nIEOuRFiUs0ZR248fD2We7XmWvvgoffAAVKwYdlTEFl5AAb73lblMzZ47bvkePDjqqUs2u0ZRS5/7+I/7y32H0XTGLlYlJPHLFAyw/zYb/MSVLo11b+MeEFzln2xomN+vMX7v9lm//fXvQYUWVorhGY7+jKW2OHoVXXmHmsL9Q7thRXupyE6917Mex2PigIzOm0K2vXpdrb/47d88bw6A5o+i6fgE02A4PPuhqPqZIWNNZaZGRAR9+CM2bwyOPsLBuc3rd9k/+df4ASzKmRMuIjePVzjdw6W9f46tGbeHxx6FJEzf8+NGjQYdXKliiKekOHYJXXoGmTeHmm6FKFZg2jdut67IpZbZUqcU9V/+P6yzQoAHcd5/bL/7xD9izJ+jwSjRLNCXVokUwaBDUresGijr9dPjsM1i4EC6zcWRM6ZU07TBJXR7nluuHMD+zEjz4IAdr1eGDc6+ABQughFy3jibWGaCkUIWVK+HTT10PmyVLOBobz5RmnXm/7RWk1GsZdITGRKWW21K5deFE+qyYRcLxY+7GsQMGwLXXQqtWJX7cpaLoDGCJpjjbtQtmznRNAdOnw5o1bnrnzjBgAOesqcnecpUCDdGY4qLykQMsbbUfPvrI7VeqUL8+XHGFawXo0gVq1w46zEJniSYfSnyi2bULli+HpUvhu+9g3jxYvRqAg/FlmVe/FTOatGdakw78XKlGwMEaU7zV2r+TrusWcPG6FLpsWESl9MNuRtOm0LGjG2zt3HOhdetiPz5TsU80ItID+CduKOc3VfX5sPkJwHtAO2AncIOqbvDmPQ7cCRwHfq+qU3NaVrFPNAcPwrZtsHmzG2jsxx9h7VpITXU1la1bfylbqxZ06MDf91RlXoNWLKnTzHqOGeOT+OPHaLl9HedtWk77zcs5e9saTjuw68T8HeWrsK56Xdpf1gEaNYKGDd2jTh33qBTdrQrFOtGISCzwA3AZsBmYDwxQ1RUhZe4DWqvqPSLSH7haVW8QkRbAx0B74HTgC6CZqh7PbnmnnGhU3SMz85e/mZluVMnQR0bGL3+PHfvlcfQopKe7v0eOuMehQy6BHDoEBw7Avn3usXcv7N7taik7d8LPP7sy4U47jfmx1dlYrQ4/1GzADzUbsjqxIVsr1Szx7cbGRLOaB3fTcvs6mu7YyBk7N9N41xY6ZOx0J4vhypeHmjXdo0YN1/OzalX3t1IldyeOChVcufLloVw59xufrEeZMhAf7/7GxbnncXEQG/vLIyEBypYt0Hcp7j/YbA+kquo6ABEZAfQBVoSU6QM85T0fDbwiIuJNH6GqR4H1IpLqfV7hD1T/3XfQoUOhf2wkh+ITOFCmPPsTyrO3bEX2lq3InvJJ7GjZhp3lq7KjQlV+qlSTbZVqsrVSTQ6XKdiGY4zx144K1ZjVuB2zGrc7aXpCRjqn70vj9H1pJB7cTa0Du0g8uJtqh/dTbe8+qm/9kUpHD1Ip/RCVjxykXEYh/Y7nhhtgxIjC+Swf+Jlo6gKbQl5vBsKP6CfKqGqGiOwFanjT54a9t274AkRkIDDQe3lARFafQrw1gR2n8P7cHTvqHgd35+dd/sdVMBZX/lhc+VNs4/qhiAI5yciRNRk5sqDry/cf1BXrW9Co6jBgWGF8loik+F19LAiLK38srvyxuPLH4ioYP3+wuQWoH/K6njctYhkRiQOq4DoF5OW9xhhjigE/E818oKmINBKRMkB/YHxYmfHArd7zfsAMdb0TxgP9RSRBRBoBTYHvfIzVGGOMT3xrOvOuuQwCpuK6N7+tqstFZAiQoqrjgbeA972L/btwyQiv3Chcx4EM4P6cepwVkkJpgvOBxZU/Flf+WFz5Y3EVQIn5waYxxpjoZDfVNMYY4ytLNMYYY3xV6hKNiNQXkS9FZIWILBeRP3jTq4vIdBFZ4/2tFiVxPSUiW0Rksfe4oojjKisi34nIEi+up73pjURknoikishIr8NHNMT1roisD1lfbYoyrpD4YkVkkYhM9F4Hur5yiCvw9SUiG0Tke2/5Kd60QPfHHOIKdH/0YqgqIqNFZJWIrBSRTtGwvnJS6hINrnPBw6raAugI3O/d8mYw8F9VbQr813sdDXEBDFXVNt5jchHHdRS4RFXPAdoAPUSkI/CCF1cTYDfuvnTREBfAH0PW1+IijivLH4CVIa+DXl9ZwuOC6FhfF3vLz/otSND7Y3ZxQbD7I7j7R05R1ebAObj/Z7Ssr4hKXaJR1a2qutB7vh/3T6qLu+3NcK/YcKBvlMQVKHUOeC/jvYcCl+BuGwTBrK/s4gqciNQDegFveq+FgNdXpLiiXKD7Y7QSkSrAhbgeu6hquqruIcrXV6lLNKFEJAk4F5gH1FbVrFskbwMCG3giLC6AQSKyVETeDqgJIVZEFgM/A9OBtcAeVc3wikS8RVBRx6WqWevrWW99DRV3h/Ci9g/gUSDTe12DKFhfEeLKEvT6UmCaiCwQd1spiI79MVJcEOz+2AhIA97xmkDfFJEKRMf6ylapTTQiUhEYAzygqvtC53k/Gg3k7DhCXK8BZ+Cah7YC/1fUManqcVVtg7tDQ3ugeVHHEEl4XCLSCngcF995QHXgsaKMSUSuBH5W1QVFudzc5BBXoOvL00VV2wI9cU3GF4bODHB/jBRX0PtjHNAWeE1VzwUOEtZMFuTxKzulMtGISDzuYP6hqn7qTd4uInW8+XVwZ8mBx6Wq270DaibwBu5AHwiviv4l0AmoKu62QRDwLYJC4urhNUGqd+fvdyj69XU+0FtENgAjcE1m/yT49fWruETkgyhYX6jqFu/vz8BYL4bA98dIcUXB/rgZ2BxSex+NSzyBr6+clLpE47WXvwWsVNWXQmaF3g7nVuCzaIgra+PxXA0sK+K4EkWkqve8HG58oZW4A3s/r1gQ6ytSXKtCdjbBtVMX6fpS1cdVtZ6qJuHudDFDVW8i4PWVTVw3B72+RKSCiFTKeg5092IIen+MGFfQ+6OqbgM2iciZ3qRuuDuoBLq+clOs795cQOcDtwDfe+37AE8AzwOjROROYCNwfZTENcDrcqrABuDuIo6rDjBc3EB2McAoVZ0oIiuAESLyV2AR3sXJKIhrhogkAgIsBu4p4riy8xjBrq/sfBjw+qoNjHV5jjjgI1WdIiLzCXZ/zC6u9wPeHwF+h/u/lQHWAbfj7QMBrq8c2S1ojDHG+KrUNZ0ZY4wpWpZojDHG+MoSjTHGGF9ZojHGGOMrSzTGGGN8ZYnGGGOMryzRGGOM8dX/A6UukhUw9pTXAAAAAElFTkSuQmCC\n", + "text/plain": [ + "<Figure size 432x288 with 1 Axes>" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "x = np.linspace(20,60,100)\n", + "plt.hist(binom_samples, bins=40, density=True)\n", + "plt.plot(x, norm(loc=40.5, scale=5).pdf(x), color=\"red\")\n", + "plt.ylabel('Dichte')\n", + "plt.title('Normalverteilungs-Dichte (rot) & Binomial-Histogramm (blau)')\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "1b266dac-1d5a-4c47-8b0a-2b08ea2fad78", + "metadata": {}, + "source": [ + ":::{admonition} Proposition\n", + "Wenn $X \\sim \\mathcal{N}(\\mu,\\sigma^2)$, so ist $\\mathbb{E}(X) = \\mu$ und $\\mathbb{V}(X) = \\sigma^2$.\n", + ":::\n", + "\n", + ":::{admonition} Beweis\n", + "Wir setzen die Dichte für $X$ in die Formel für $\\mathbb{E}(X)$ ein und transformieren das Integral über $x$ in ein Integral über $x + \\mu$:\n", + "\n", + "$$\n", + "\\mathbb{E}(X) &= \\int_{-\\infty}^{\\infty} x\\phi(x) dx \\\\\n", + " &= \\int_{-\\infty}^{\\infty} x \\frac{1}{\\sigma\\sqrt{2\\pi}} e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}} dx \\\\\n", + " &= \\frac{1}{\\sigma\\sqrt{2\\pi}} \\int_{-\\infty}^{\\infty} x e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}} dx \\\\\n", + " &= \\frac{1}{\\sigma\\sqrt{2\\pi}} \\int_{-\\infty}^{\\infty} (x+\\mu) e^{-\\frac{x^2}{2\\sigma^2}} dx \\\\\n", + " &= \\frac{1}{\\sigma\\sqrt{2\\pi}} \\left( \\left( \\int_{-\\infty}^{\\infty} x e^{-\\frac{x^2}{2\\sigma^2}} dx \\right) + \\mu \\left( \\int_{-\\infty}^{\\infty} e^{-\\frac{x^2}{2\\sigma^2}} dx \\right) \\right) \\\\\n", + " &= \\frac{1}{\\sigma\\sqrt{2\\pi}} \\left( \\left( \\int_{-\\infty}^{\\infty} x e^{-\\frac{x^2}{2\\sigma^2}} dx \\right) + \\mu \\sigma\\sqrt{2\\pi} \\right) \\\\\n", + " &= \\mu + \\frac{1}{\\sigma\\sqrt{2\\pi}} \\left( \\int_{-\\infty}^{\\infty} x e^{-\\frac{x^2}{2\\sigma^2}} dx \\right) = \\mu\\\\\n", + "$$\n", + "\n", + "Der Ausdruck nach dem $\\mu +$ ist einfach $0$, denn die Funktion ist punktsymmetrisch um $0$ (der Faktor $x$ ist offensichtlich eine ungerade Funktion, der andere Faktor hängt nur von $|x|$ ab).\n", + "\n", + "Genau so kann man bei der Varianz verfahren, indem man diesen Ausdruck vereinfacht:\n", + "\n", + "$$\n", + "\\mathbb{V}(X) &= \\int_{-\\infty}^{\\infty} x^2\\phi(x) dx - {\\left( \\mathbb{E}(X) \\right)}^2 \\\\\n", + " &= \\int_{-\\infty}^{\\infty} x^2\\frac{1}{\\sigma\\sqrt{2\\pi}} e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}} dx - \\mu^2\n", + "$$\n", + ":::" + ] + }, + { + "cell_type": "markdown", + "id": "e67caa4e-c0d6-4f44-a9e8-620dc7b20bf3", + "metadata": {}, + "source": [ + ":::{admonition} Definition\n", + "Man nennt eine Zufallsvariable mit $X \\sim \\mathcal{N}(0,1)$ auch *standardnormalverteilt*\n", + ":::\n", + "\n", + ":::{admonition} Proposition\n", + "Wenn $X \\sim \\mathcal{N}(\\mu,\\sigma^2)$, so ist $\\frac{X - \\mu}{\\sigma} \\sim \\mathcal{N}(0,1)$.\n", + ":::\n", + "\n", + "\n", + ":::{admonition} Beweis\n", + "**Idee:**\n", + "Es ist sofort klar, dass $\\mathbb{E}(X-\\mu) = 0$ und dass $\\mathbb{V}\\left(\\frac{X-\\mu}{\\sigma}\\right) = 1$.\n", + "Weniger klar ist, dass die neue Zufallsvariable tatsächlich normalverteilt ist. Das lässt sich z.B. mit der Momenterzeugendenfunktion beweisen.\n", + ":::\n", + "\n", + "Alternativ nutzen wir, dass die Summe von unabhängigen Zufallsvariablen mit Dichten $f,g$ selbst wieder eine Dichte hat, nämlich die *Faltung* $f \\ast g$. Anstatt das nun rigoros einzuführen und zu beweisen, benutzen wir es ein weiteres Mal, damit lässt sich nämlich zeigen:\n", + "\n", + ":::{admonition} Proposition\n", + "Wenn $X \\sim \\mathcal{N}(\\mu_1,\\sigma_1^2)$ und $Y \\sim \\mathcal{N}(\\mu_2,\\sigma_2^2)$,\n", + "so ist $X+Y \\sim \\mathcal{N}(\\mu_1 + \\mu_2,\\sigma_1^2 + \\sigma_2^2)$.\n", + ":::\n", + "\n", + ":::{admonition} Satz\n", + "Unter allen Verteilungen reeller Zufallsvariablen $X$ mit festem Erwartungswert $\\mu = \\mathbb{E}(X)$ und Varianz $\\sigma^2 = \\mathbb{V}(X)$ ist die Normalverteilung diejenige mit der maximalen Entropie.\n", + ":::\n", + "\n", + "Um diesen Satz überhaupt präzise formulieren zu können, benötigen wir einen Entropiebegriff für stetige Verteilungen. Anstatt das jetzt zu tun, wollen wir uns später damit beschäftigen, wenn wir auch relative Entropie, die stetige Version davon und Likelihood diskutieren.\n", + "\n", + "Wichtig ist aber die Take-Home-Message des Satzes: Wenn über eine stetige Verteilung reeller Zahlen außer Erwartungswert und Varianz nichts bekannt ist, dann ist die entprechende Normalverteilung die vernünftigste Annahme. In diesem Sinne ist die Normalverteilung ein guter stetiger Ersatz für die diskrete Gleichverteilung, aber nun auf ganz $\\mathbb{R}$ (die Normalverteilung hat Träger $\\mathbb{R}$)." + ] + }, + { + "cell_type": "markdown", + "id": "29d32c3a-91c9-4534-adfa-d0fd87a7680d", + "metadata": {}, + "source": [ + "# Der zentrale Grenzwertsatz\n", + "\n", + ":::{admonition} Satz\n", + "Wenn $X_i$ eine Folge von identisch verteilten, voneinander unabhängigen (iid = independent identically distributed) Zufallsvariablen mit Erwartungswert $\\mathbb{E}(X_i)=0$ und Varianz $\\mathbb{V}(X_i)=\\sigma^2$ ist,\n", + "und wir die skalierte Summe $S_N := \\dfrac{\\sqrt{N}}{N} \\sum_{i=1}^N X_i$ betrachten, dann gilt die Konvergenz in Verteilung:\n", + "\n", + "$$\n", + "S_N \\xrightarrow{N \\to \\infty} \\mathcal{N}(0,1)\n", + "$$\n", + ":::\n", + "\n", + "Konvergenz in Verteilung heißt, dass die Verteilungsfunktion von $S_N$ gegen die Verteilungsfunktion einer Standardnormalverteilung konvergiert.\n", + "\n", + "Von diesem Satz gibt es auch Abschwächungen, die gewisse Abhängigkeiten zwischen den $X_i$ erlauben.\n", + "\n", + ":::{admonition} Bemerkung\n", + "Wichtig ist der Satz für uns, weil er erlaubt eine Summe unabhängiger Zufallsvariablen mit einer Normalverteilung zu approximieren. Das tritt in der Praxis häufig auf, wenn man Messungen an physikalischen oder technosozialen Systemen vornimmt, die in der Regel von einer langen Liste weitgehend unabhängiger Störeinflüsse beeinträchtigt werden. Diese Summe an Fehlerquellen ist (für eine hinreichend große Zahl an unabhängigen Fehlerquellen) etwa normalverteilt.\n", + ":::\n", + "\n", + "Um zwei konkrete Beispiele zu nennen: der Fehler bei der Ortsbestimmung mit GNSS-Systemen wie GPS ist normalverteilt (auch wenn die atmosphärischen Störungen, die den Wert ungenauer machen, nicht normalverteilt sind). Bei der industriellen Fertigung von Bauteilen gibt es ebenfalls im gesamten Produktionsprozess Fehlerquellen, die am Ende zu einer Normalverteilung aufsummieren - deren Varianz man hinreichend klein halten muss, damit das Bauteil seine Aufgabe erfüllen kann. So darf ein Legostein nicht zu stark von einem baugleichen Legostein abweichen, sonst hält das Bauwerk hinterher nicht richtig." + ] + }, + { + "cell_type": "markdown", + "id": "961bb23c-049c-4367-866c-a840b6fab2e8", + "metadata": {}, + "source": [ + "Man kann außerdem die Geschwindigkeit der Konvergenz abschätzen, und damit in Erfahrung bringen, wie gut die Approximation durch eine Normalverteilung ist:\n", + "\n", + ":::{admonition} Satz\n", + "Es existiert eine Konstante $C > 0,4409$ mit der folgenden Eigenschaft:\n", + "\n", + "Wenn $X_i$ eine Folge von iid Zufallsvariablen mit Erwartungswert $0$ und Varianz $\\sigma^2$ ist, und außerdem die dritten absoluten Momente $\\mathbb{E}|X_i^3| = \\rho < \\infty$ existieren,\n", + "wir $F_N$ für die kumulative Verteilungsfunktion von $\\frac{\\sqrt{N}}{N\\sigma} \\sum_{i=1}^N X_i$ schreiben und $\\Phi$ für die kumulative Verteilungsfunktion der Standardnormalverteilung $\\mathcal{N}(0,1)$, so gilt für alle $x$ und alle $N$:\n", + "\n", + "$$\n", + "\\left| F_n(x) - \\Phi(x) \\right| \\ \\leq\\ \\dfrac{C\\rho}{\\sigma^3 \\sqrt{n}}.\n", + "$$\n", + ":::\n", + "\n", + "Für konkret bekannte $\\sigma,\\rho$ können wir also stets das $n$ bestimmen, um die linke Seite beliebig klein abzuschätzen. Mit anderen Worten: es lässt sich zu jeder gewünschten Genauigkeit berechnen, wie viele der iid Zufallsvariablen man (skaliert) aufaddieren muss, um es mit einer Standardnormalverteilung zu approximieren.\n", + "\n", + "Klar: Wenn der Erwartungswert nicht $0$ ist, lässt sich eine Variante als Korollar beweisen (wie auch beim zentralen Grenzwertsatz), sodass die Konvergenz gegen $\\mathcal{N}(\\mu,1)$ geht. Wenn man anders reskaliert, auch gegen $\\mathcal{N}(\\mu,\\sigma^2)$. In der Praxis sieht es eher so aus, dass man alle unbekannten Fehlerterme mit einer Normalverteilung modelliert, deren Erwartungswert und Varianz man empirisch schätzt.\n" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.8.10" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/stetige-verteilungen.md b/stetige-verteilungen.md new file mode 100644 index 0000000000000000000000000000000000000000..cef788f656ef043551912a38b6f229f6123a8057 --- /dev/null +++ b/stetige-verteilungen.md @@ -0,0 +1,105 @@ +# Stetige Verteilungen + +:::{admonition} Definition +Sei $X \colon \Omega \to \mathbb{R}$ eine Zufallsvariable, d.h. $\Omega$ ein Wahrscheinlichkeitsraum (eine Menge $\Omega$ zusammen mit einem Wahrscheinlichkeitsmaß $P_\Omega$) und $X$ eine Abbildung, sodass $P(X \in A) := P_\Omega(X^{-1}(A))$ für alle Ereignisse $A \subseteq \mathbb{R}$ definiert ist. + +Wenn $\Omega$ kein diskreter Wahrscheinlichkeitsraum ist (z.B. weil $\Omega$ eine überabzählbar unendliche Menge ist, etwa $\Omega = \mathbb{R}^n$), und $X$ unendlich viele mögliche Werte annimmt, nennen wir $X$ eine *stetige Zufallsvariable*. +::: + +:::{admonition} Definition +Man nennt die kleinste Menge $A \subset \mathbb{R}$ mit $P(X \in A)=1$ den *Träger* von $X$ und schreibt auch + +$$ +supp(X) = \bigcap \{ A \subseteq \mathbb{R} : P(X \in A) = 1 \} +$$ +::: + +Der Träger ist auch für diskrete reelle Zufallsvariablen definiert - der offensichtliche Unterschied ist, dass per definitionem diskrete Zufallsvariablen einen diskreten Träger haben (d.h. eine Menge reeller Zahlen, die keinen Häufungspunkt hat, insbesondere kein Intervall enthält). + +## Dichtefunktionen + +:::{admonition} Definition +Sei $\Omega \subset \mathbb{R}^n$ und $f \colon \Omega \to [0,\infty)$ eine integrierbare Funktion (bezüglich des Lebesgue-Maßes auf $\mathbb{R}^n$, Riemann-integrierbar ist ein hinreichendes Kriterium) mit $\int_\Omega f(x) dx = 1$. Dann ist auf $\Omega$ ein Wahrscheinlichkeitsmaß definiert durch: + +$$ +\text{für } A \subseteq \Omega \text{ Ereignis } P(A) := \int_A f(x) dx +$$ + +Wir nennen für jedes Wahrscheinlichkeitsmaß $P$, welches sich so schreiben lässt, die Funktion $f$ eine *Wahrscheinlichkeitsdichtefunktion* (probability density function, pdf) oder kurz *Dichte* (gelegentlich Wahrscheinlichkeitsmassefunktion, pmf). +::: + +Gegeben eine reelle Zufallsvariable $X \colon \Omega \to \mathbb{R}$, deren Verteilung $P_X$ durch eine Dichte $f$ gegeben ist, gilt also für $A \subseteq \mathbb{R}$ + +$$ +\int_{X^{-1}(A)} d\omega = P(X \in A) = P_X(A) = \int_A f(x) dx +$$ + +Dieses rechte Integral ist nun ein Integral im Wertebereich von $X$. +Analog können wir auch den Erwartungswert und die Varianz berechnen: + +$$ +\mathbb{E}(X) = \int_\Omega X(\omega) d \omega = \int_{\mathbb{R}} x f(x) dx = \int_{-\infty}^{\infty} xf(x) dx +$$ + +$$ +\mathbb{V}(X) &= \int_\Omega \left(X(\omega)-\mathbb{E}(X)\right)^2 d \omega \\ + &= \int_{-\infty}^{\infty} \left(x-\mathbb{E}(X)\right)^2 f(x) dx \\ + &= \int_{-\infty}^{\infty} x^2f(x) dx - {\left( \mathbb{E}(X) \right)}^2 +$$ + +:::{admonition} Beispiel +Seien $a < b \in \mathbb{R}$. Mit $\Omega = \mathbb{R}^1$ und $f \colon \mathbb{R} \to [0,\infty)$ gegeben durch + +$$ +f(x) = \begin{cases} \frac{1}{b-a}, & \text{ wenn } x \in [a,b] \\ 0 & \text{ sonst} \end{cases} +$$ + +ist ein Wahrscheinlichkeitsmaß auf $\mathbb{R}$ definiert, die *stetige Gleichverteilung* auf dem Intervall $[a,b]$. +Der Erwartungswert (wir nutzen $\int x dx = \frac{x^2}{2}$) ist $\mathbb{E}(X) =$ + +$$ +\int_{-\infty}^{\infty} xf(x) dx = \int_a^b \frac{xdx}{b-a} = \frac{1}{b-a} \left[\frac{x^2}{2}\right]_a^b = \frac{b^2 -a^2}{2(b-a)} = \frac{(b+a)(b-a)}{2(b-a)} = \frac{a+b}{2} +$$ + +Das zweite Moment ist + +$$ +\mathbb{E}(X^2) = \int_{-\infty}^\infty x^2 f(x)dx = \left[ \frac{x^3}{3(b-a)} \right]_a^b = \frac{b^3-a^3}{3(b-a)} = \frac{a^2+ab+b^2}{3} +$$ + +Die Varianz ist + +$$ +\mathbb{V}(X) = \mathbb{E}(X^2) - {\left(\mathbb{E}(X)\right)}^2 = \frac{a^2+ab+b^2}{3} - \frac{(a+b)^2}{4} = \frac{(a-b)^2}{12} +$$ +::: + +:::{admonition} Beispiel +Nicht jedes Wahrscheinlichkeitsmaß erlaubt eine Darstellung mit einer Dichte: +Betrachte auf $\mathbb{R}$ die Verteilung mit + +$$ +P(A) = \begin{cases} 1 & \text{ wenn } 0 \in A \\ 0 & \text{ sonst} \end{cases} +$$ + +Diese Verteilung heißt *Dirac-Verteilung* mit Masse bei $0$. Wenn man sich sehr viel Mühe gibt, so etwas ähnliches wie eine Dichte zu basteln, so muss man zur Theorie der Distributionen aus der Funktionalanalysis greifen (Physiker kennen das). Es gibt auch Verteilungen, da genügt auch keine Distribution. +::: + +:::{admonition} Definition +Sei $(\Omega, P)$ ein Wahrscheinlichkeitsraum, dann nennen wir für eine Zufallsvariable $X \colon \Omega \to \mathbb{R}$x die Funktion + +$$ +F \colon \mathbb{R} \to [0,1],\qquad x \mapsto P(X \leq x) +$$ + +die *kumulative Wahrscheinlichkeitsverteilungsfunktion* oder auch *Verteilungsfunktion* oder *cdf* (cumulative distribution function). Sie existiert immer, im Gegensatz zur Dichte (pdf). +::: + +:::{admonition} Beispiel +Wenn $P_X$ eine Dichte $f \colon \mathbb{R} \to [0,\infty)$ hat, +also $P(a \leq X \leq b) = \int_a^b f(x)dx$, so ist + +$$ +F(x) = \int_{-\infty}^x f(x)dx. +$$ +::: diff --git a/wichtigste-stetige-verteilungen.ipynb b/wichtigste-stetige-verteilungen.ipynb new file mode 100644 index 0000000000000000000000000000000000000000..2d1fe538054dd3dbbb44decc2e34dad6544fa0f8 --- /dev/null +++ b/wichtigste-stetige-verteilungen.ipynb @@ -0,0 +1,784 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "fcbdd137-53b6-4407-9db8-7583a9e04a91", + "metadata": {}, + "source": [ + "# Wichtigste stetige Verteilungen\n", + "\n", + "Nachdem wir ausführlich über die Gaussverteilung gesprochen haben, wollen wir einen Überblick über einige der wichtigsten stetigen Verteilungen bekommen, ähnlich wie wir ihn bereits für diskrete Verteilungen gewonnen haben. Dazu benutzen wir das Python-Modul `scipy.stats`." + ] + }, + { + "cell_type": "markdown", + "id": "91efdf3d-a441-4b69-84b8-614c57e29986", + "metadata": {}, + "source": [ + "## SciPy Statistics" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "id": "fe446f00-6f77-459e-a994-340e80d88445", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "some examples: [4.02801895 1.9033209 6.49477125 1.83362523]\n" + ] + } + ], + "source": [ + "import numpy as np\n", + "np.random.seed(12321)\n", + "\n", + "import scipy.stats as st\n", + "\n", + "# Zufallsvariable mit N(3,2^2)-Verteilung erzeugen:\n", + "X = st.norm(loc=3, scale=2)\n", + "# samples ziehen:\n", + "Xsamples = X.rvs(size=4)\n", + "print(\"some examples:\",Xsamples)" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "dedb9580-1b87-4e59-917e-ca0865c0c379", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "mean: 3.0004822746684243 expected 3.0\n", + "std: 1.9994507762900595 expected 2.0\n", + "3rd moment: 62.99999999999999\n", + "4th moment: 344.99999999999994\n", + "entropy: 2.112085713764618\n" + ] + } + ], + "source": [ + "# Verteilung checken:\n", + "Xsamples = X.rvs(size=1000)\n", + "print(\"mean:\", np.mean(Xsamples), \"expected\", X.mean()) # 3\n", + "print(\"std:\", np.std(Xsamples), \"expected\", X.std()) # 2\n", + "print(\"3rd moment:\", X.moment(3))\n", + "print(\"4th moment:\", X.moment(4))\n", + "print(\"entropy:\", X.entropy())" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "id": "0bbb8d6a-9cae-48a2-a197-ee6ff8458fdf", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "3.0004822746684243 1.9994507762900595\n" + ] + } + ], + "source": [ + "# Daten fitten:\n", + "mu, sigma = st.norm.fit(Xsamples)\n", + "print(mu, sigma)" + ] + }, + { + "cell_type": "markdown", + "id": "e51c0170-cdf2-4b52-a1b7-dd240895caab", + "metadata": {}, + "source": [ + "Es gibt noch weitere praktische Methoden, etwa `st.norm.sf`, die 'survival function', definiert als 1-cdf oder `st.norm.logpdf`, falls man direkt den Logarithmus der Dichtefunktion verwenden möchte (beide Methoden sind potentiell schneller, als das von Hand / zu Fuß auszurechnen)." + ] + }, + { + "cell_type": "markdown", + "id": "78b5821b-84e5-468c-a368-7b93c22f8117", + "metadata": {}, + "source": [ + "## Matplotlib\n", + "\n", + "Wir werden uns von den Verteilungen jeweils Plots der Dichtefunktionen (pdf) und der kumulierten Verteilungsfunktionen (cdf) anschauen. Dazu verwenden wir das Python-Modul `matplotlib.pyplot`. Es emuliert das Verhalten von MATLAB. Wir werden uns noch ausführlicher mit Matplotlib beschäftigen." + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "id": "2ce85da0-8c6f-4f83-9339-195fda4982bf", + "metadata": {}, + "outputs": [], + "source": [ + "from matplotlib import pyplot as plt" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "id": "76221be7-043b-41e2-93a8-ae908de80132", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "sample[:10] = [5.96301841 5.60942173 5.72908826 5.80068973 5.59819708 5.35377228\n", + " 6.093922 6.36189002 5.42430139 5.87334923]\n" + ] + }, + { + "data": { + "image/png": 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\n", + "text/plain": [ + "<Figure size 432x288 with 1 Axes>" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + }, + { + "data": { + "image/png": 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\n", + "text/plain": [ + "<Figure size 432x288 with 1 Axes>" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + }, + { + "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": [ + "# Größe der Plotausgabe anpassen:\n", + "plt.rcParams[\"figure.figsize\"] = (16,9)\n", + "\n", + "# Sample der Verteilung von einer früheren Übungsaufgabe:\n", + "sample = np.random.choice(3, 10000, p=[4/7,2/7,1/7]) + 5 + np.random.random(10000)\n", + "print(\"sample[:10] =\", sample[:10])\n", + "# Histogramm plotten:\n", + "plt.hist(sample, bins=100)\n", + "plt.ylabel(\"Häufigkeit\")\n", + "plt.xlabel(\"Mittelwert=\"+str(np.mean(sample))+\" (erwartet: \"+str(6+1/14)+\"), Standardabweichung²=\"+str(np.std(sample)**2))\n", + "plt.show()\n", + "\n", + "plt.hist(sample, bins=100, density=True)\n", + "plt.ylabel(\"Dichte\")\n", + "plt.xlabel(\"Mittelwert=\"+str(np.mean(sample))+\" (erwartet: \"+str(6+1/14)+\"), Standardabweichung²=\"+str(np.std(sample)**2))\n", + "plt.show()\n", + "\n", + "# Histogramm mit wenig bins:\n", + "plt.hist(sample, bins=3)\n", + "plt.ylabel(\"Häufigkeit\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "6b91accc-856e-46b0-8d19-e91af14aeda0", + "metadata": {}, + "source": [ + "Zum Plotten einer Funktion definieren wir diese als Python-Methode und rechnen hinreichend viele Werte aus, zwischen denen dann interpoliert wird:" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "id": "fd4df7ae-888f-4f8b-aad5-794ebdc870ac", + "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": [ + "# Funktion x |-> 3+2x+x²-x³ auf Intervall [-80,90] plotten:\n", + "def f(x):\n", + " return 3 + 2*x + x**2 - x**3\n", + "\n", + "x = np.linspace(-80, 90, 100)\n", + "plt.plot(x, f(x), color='red')\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "c9a1de69-9590-4ef4-b708-c202dfda6d50", + "metadata": { + "tags": [] + }, + "source": [ + "Damit kümmern wir uns nun um die wichtigsten stetigen Verteilungen:\n", + "\n", + "## Normalverteilung\n", + "\n", + "* Parameter $\\mu,\\sigma$, Notation $\\mathcal{N}(\\mu,\\sigma^2)$\n", + "* Dichte $\\phi(x) = \\frac{1}{\\sigma\\sqrt{2\\pi}} e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}}$\n", + "* Erwartungswert $\\mu$\n", + "* Varianz $\\sigma^2$\n", + "* Höhere ungerade zentrale Momente sind alle $0$ \\\n", + " (daher sind ungerade Momente ein Maß für nicht-Normalität)\n", + "* Maximale Entropie bei festem Erwartungswert und fester Varianz und Träger ganz $\\mathbb{R}$\n", + "* ZGWS: Folge iid Zufallsvariablen $X_i$, dann $\\sum_{i=1}^N X_i \\xrightarrow{N \\to \\infty} \\mathcal{N}(\\mu,\\sigma^2)$\n", + "* Anwendungsbeispiele:\n", + " * Messfehler in physikalischen Systemen\n", + " * Fertigungsgenauigkeit\n", + " * Biologische Größen (Körperlänge etc.)" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "id": "334bb8a6-1046-454b-ac1c-9b921d5136b4", + "metadata": { + "tags": [] + }, + "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" + }, + { + "data": { + "image/png": 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cfGgvjhnajcy01KhDbDWUCESkxSirqOLVJZt4dPYa3luxjdQU47ihOfzylBEcP7w7Wek6+MeDEoGIRK6guJxH3l/N395dxZbCUnp3asPPvzWMs3P70L19VtThtXpKBCISmU0FJdz75nIen7OW4rJKjhnSjdvPHMmxQ3NITdFVPs1FiUBEmt3Ggj3837+WM2POWiqrnImjDuCSYwYx4oAOUYeWlJQIRKTZ7Cwu48//Ws6D766iqso5O7cP/3ncYPp2aRt1aElNiUBE4q60opIH31nFPW/kUVhawZlH9OGy44coAbQQSgQiEldvfLKFm19cwsr83Rw3LIdrTjqIg3qqCaglUSIQkbhYt6OYm2Yu5rWlWxjULZsHfzCa44Z1jzosqYUSgYg0qcoq56H3VvGbVz4F4NqTDuIHRw0kIy2xb9XcmikRiEiTWb61iJ89+THz1+zkG0NzuO2MQ+jTWf0ALZ0SgYjsN3fn4dmr+dWspWSlp/KHcw9j4mEH6I6fCUKJQET2y5bCEq584mPeWpbPN4bmcMdZI+nRQb8GTiRKBCLSaG8vy+fyx+dTVFrBracfwgVj+6kWkICUCERkn1VWOXe9vow//XMZB+a047HJ4xjao33UYUkjKRGIyD7ZWVzGT6fP561l+Zx5RB9uOf1gDfmY4LT1RKTBPtm0iykPzWNjwR5+dcahnD+2X9QhSRNQIhCRBnl18SYuf/wjsjPTmDFlHEf27xJ1SNJElAhEpE7uzv1vr+S2WUsZ2bsj0y7M1VVBrYwSgYjsVUVlFf/zwhIenr2aCQf35PfnHkabDI0S1tooEYhIrUrKK5n62HxeW7qZHx07iKsnHESKBotplZQIROQrdpWUc8nf5jJn1XZunngwF35tQNQhSRwpEYjIl2wpLOH7D8whb0shfzzvcE4ddUDUIUmcKRGIyOc2FZRw/n2z2VhQwv3fH82xQ3OiDkmagRKBiADB+AHn3/c+23eX8fDFY8gdoMtDk4USgYiwZlsxk+6bza6Sch6+eAyH9+scdUjSjJQIRJLcuh1BEthdVsH0yeM4pHfHqEOSZqYhg0SS2MaCPZx/3/sUlpTzyMVjlQSSVFwTgZlNMLNPzSzPzK6p5fl+ZvaGmc03swVmdnI84xGRL2wpLOGCsE/gISWBpBa3RGBmqcA9wEnACGCSmY2oUex64Al3Pxw4D/hzvOIRkS8UFJdz4f0fsGlXCQ/+YDSH9e0UdUgSoXjWCMYAee6+wt3LgBnAxBplHOgQ/t8R2BDHeEQE2FNWyQ//NocVW3dz34W5ujpI4tpZ3BtYGzO9Dhhbo8xNwKtm9lMgGzghjvGIJL3yyip+8ug85q/ZwT3nH8FRg7tFHZK0AFF3Fk8CHnT3PsDJwMNm9pWYzGyKmc01s7lbt25t9iBFWgN35+qnFvCvT7dy2xmHctKhvaIOSVqIeCaC9UDfmOk+4bxYFwNPALj7e0AW8JVTFHef5u657p6bk6NfOoo0xu9e/Yxn5q/nyvFDmTRGA8rIF+KZCOYAQ8xsoJllEHQGz6xRZg1wPICZDSdIBDrlF2li0z9Yw91v5HHe6L5M/ebgqMORFiZuicDdK4CpwCvAUoKrgxab2c1mdlpY7Epgspl9DEwHLnJ3j1dMIsnojU+3cP1zi/jG0BxuOf0QzHQrafmyuP6y2N1nAbNqzLsh5v8lwFHxjEEkmX2yaRdTH/2QYT3ac88FR5CeGnW3oLRE2itEWqn8olIufnAu2ZlpPHDRaNpl6o4yUjvtGSKtUGlFJT96eB75RaU8+eOv0bOjxhiWvVMiEGll3J1rn1nIvNU7uPv8wxnZp1PUIUkLp6YhkVbmgXdW8cyH67n8hCGcMlKji0n9lAhEWpF3l+fzq1lLOXFEDy795pCow5EEoUQg0kqs21HM1MfmM6hbNneeexgpKbpMVBpGiUCkFSgpr+THj8yjvLKKaRfm6goh2SfaW0RagRufX8yi9bt44KJcBnbLjjocSTCqEYgkuCfmrOXxuWuZ+h+D+eZBPaIORxKQEoFIAlu0voBfPr+IowZ35b/HD406HElQSgQiCWpXSTn/+eiHdG6bwV3nHU6qOoelkdRHIJKA3J1rnl7A+p17eHzKOLq1y4w6JElgqhGIJKBH3l/DrIWb+Pm3hmmoSdlvSgQiCWbxhgJueXEJxw3LYcoxg6IOR1oBJQKRBFJUWsHUx+bTuW06vzt7lH40Jk1CfQQiCeTG5xezettupk8eR1f1C0gTUY1AJEE8/9F6nv5wHVO/OYSxg7pGHY60IkoEIglg7fZirn92EUf278ylGnNYmpgSgUgLV1FZxaUz5oPBH849jDQNNylNTH0EIi3cn/6Zx/w1O/njpMPp26Vt1OFIK6RTC5EW7MM1O7j7jTy+c3hvThulQWYkPpQIRFqo3aUV/PfjH9GzQxY3TTw46nCkFVPTkEgLdcuLS1izvZjHp3yNDlnpUYcjrZhqBCIt0D+WbGbGnLX85BsHMmagbiEh8aVEINLCbCsq5dpnFjCiVwcuP0G3lpb4U9OQSAvi7lz37EJ27ang0UsOIyNN52oSf9rLRFqQZ+ev55XFm7nixKEM69k+6nAkSSgRiLQQG3bu4caZi8nt35nJuquoNCMlApEWwN25+ukFVFQ6vztnlEYbk2alRCDSAsyYs5a3luVz3ckH0b9rdtThSJJRIhCJ2Lodxdz64hK+fmBXLhjbP+pwJAnFNRGY2QQz+9TM8szsmr2UOcfMlpjZYjN7LJ7xiLQ0VVXOVU8tAOD2M0dqoBmJRNwuHzWzVOAeYDywDphjZjPdfUlMmSHAtcBR7r7DzLrHKx6RlujRD9bw7vJt/OqMQ3VDOYlMPGsEY4A8d1/h7mXADGBijTKTgXvcfQeAu2+JYzwiLcq6HcX8etZSjh7cjUlj+kYdjiSxeCaC3sDamOl14bxYQ4GhZvaOmc02swm1vZGZTTGzuWY2d+vWrXEKV6T5uDvXPrMQgF+feShmahKS6ETdWZwGDAGOAyYB95lZp5qF3H2au+e6e25OTk7zRigSB4+HVwldc/Jw+nRWk5BEK56JYD0QW9/tE86LtQ6Y6e7l7r4S+IwgMYi0WhsL9nDbS0sZN6gLF4zpF3U4InFNBHOAIWY20MwygPOAmTXKPEdQG8DMuhE0Fa2IY0wikXJ3rntmIRVVzh1njtJVQtIixC0RuHsFMBV4BVgKPOHui83sZjM7LSz2CrDNzJYAbwA/d/dt8YpJJGrPfbSeNz7dys+/NYx+XdUkJC2DuXvUMeyT3Nxcnzt3btRhiOyzrYWljP/9vxnULZsnf/x13UZCmpWZzXP33Nqei7qzWCRp3DRzMcWlldxx1kglAWlRGpQIzGyomb1uZovC6ZFmdn18QxNpPf6+aBMvLdzIZScMYXB33V5aWpaG1gjuI/gFcDmAuy8g6PwVkXoUFJfzy+cXMaJXB6Ycq9tLS8vT0ETQ1t0/qDGvoqmDEWmNfjVrKdt3l3HHWSNJT1VrrLQ8Dd0r883sQMABzOwsYGPcohJpJd7Jy+fxuWuZfMwgDundMepwRGrV0JvO/RcwDTjIzNYDK4EL4haVSCuwp6ySa59ZyMBu2Vx+gn4nKS1XQxOBu/sJZpYNpLh7oZkNjGdgIonuzn98yprtxcyYMo6s9NSowxHZq4Y2DT0N4O673b0wnPdUfEISSXwfr93J/W+vZNKYfowb1DXqcETqVGeNwMwOAg4GOprZd2Ke6gBkxTMwkURVXlnF1U8vIKd9JteefFDU4YjUq76moWHAKUAn4NSY+YUEYwmISA3T3lzBJ5sKmfa9I+mQlR51OCL1qjMRuPvzwPNm9jV3f6+ZYhJJWMu3FnHX68v49qG9OPHgnlGHI9IgDe0szjOz64ABsa9x9x/GIyiRRFRV5Vz79ELapKdy42kjog5HpMEamgieB94CXgMq4xeOSOKaPmcNH6zazh1njqR7e3WhSeJoaCJo6+5XxzUSkQS2qaCEX8/6hK8f2JWzc/tEHY7IPmno5aMvmtnJcY1EJEG5O9c/t4iyyir+9zsaf1gST32XjxYS3FbCgOvMrJTgxnNG8COzDvEPUaRle3nRJl5buplrTzqI/l2zow5HZJ/Vd9WQ7pcrUoeC4nJueH4xh/TuwMVH68f2kpgaOh7BGWbWMWa6k5mdHreoRBLEbbOWsKO4jNvPHEma7iwqCaqhe+6N7l5QPeHuO4Eb4xKRSIJ4Jy+fJ+auY8qxgzj4AN1ZVBJXQxNBbeUaesWRSKsTe2fRy47XnUUlsTU0Ecw1szvN7MDwcScwL56BibRkv3/tM9ZsL+Z/v3Oo7iwqCa+hieCnQBnwePgoJRijQCTpLFi3k7+8tUJ3FpVWo0HNO+6+G7gmzrGItHjllVVc9ZTuLCqtS32/I/iDu19uZi8QDlMZy91Pi1tkIi3Qvf9eziebCrnvwlzdWVRajfpqBA+Hf38b70BEWrq8LYX88fU8vj2yF+NH9Ig6HJEmU98PyuaFf/9tZjnh/1ubIzCRlqSqyrnm6YW0zUzlplMPjjockSZVb2exmd1kZvnAp8BnZrbVzG6If2giLcdD761i7uod/PLbI8hpnxl1OCJNqs5EYGZXAEcBo929i7t3BsYCR5nZfzdHgCJRW7u9mDte+ZRvDM3hO0f0jjockSZXX43ge8Akd19ZPcPdVwDfBS6MZ2AiLYG7c+0zCzHgV7qzqLRS9SWCdHfPrzkz7CfQJRPS6j0xdy1v5+VzzcnD6d2pTdThiMRFfYmgrJHPiSS8TQUl3PrSUsYM7MIFY/pFHY5I3NSXCEaZ2a5aHoXAofW9uZlNMLNPzSzPzPb6gzQzO9PM3Mxy93UFROLB3fnFswspr6zi9jNHkpKiJiFpveq7fLTRN1Exs1TgHmA8sA6YY2Yz3X1JjXLtgcuA9xu7LJGm9txH63n9ky1c/+3hDOymwWakdYvnDdTHAHnuvsLdy4AZwMRayt0C3A6UxDEWkQbbUljCTTOXcES/TvzgKA02I61fPBNBb2BtzPS6cN7nzOwIoK+7v1TXG5nZFDOba2Zzt27V79kkftydXz63iD3lldxx1ihS1SQkSSCyIZXMLAW4E7iyvrLuPs3dc909NycnJ/7BSdKa+fEGXlm8mSvGD2Vw93ZRhyPSLOKZCNYDfWOm+4TzqrUHDgH+ZWargHHATHUYS1S27CrhhucXc1jfTlyi8YclicQzEcwBhpjZQDPLAM4DZlY/6e4F7t7N3Qe4+wBgNnCau8+NY0witXJ3rns2aBL67dmjNP6wJJW47e3uXgFMBV4BlgJPuPtiM7vZzHT7amlRnvtoPa8t3czPTxymJiFJOnEdd9jdZwGzasyr9YZ17n5cPGMR2ZtNBSXc+PxijuzfmR+qSUiSkOq/ktTcnaufXkBZZRW/OWukrhKSpKREIEntsQ/W8O/PtnLdycMZlKMmIUlOSgSStFZv281tLy3l6MHd+O7Y/lGHIxIZJQJJSpVVzpVPfExqinHHWbqXkCS3uHYWi7RU9765nLmrd3DnOaM4QLeXliSnGoEknUXrC7jz1c/49qG9OONwjTgmokQgSWVPWSWXzZhP13YZ3HbGIRpxTAQ1DUmS+fXLS1m+dTcPXzyGTm0zog5HpEVQjUCSxhufbuFv763mh0cN5JghunmhSDUlAkkKWwtL+fmTHzOsR3uumjAs6nBEWhQ1DUmrV1XlXPnkxxSWVPDY5HFkpTd64D2RVkk1Amn1HnhnJW9+tpXrTxnB0B7tow5HpMVRIpBWbdH6Am7/+yeMH9GD747tF3U4Ii2SEoG0WoUl5Ux97EO6Zmdy+5kjdamoyF6oj0BapeqBZtZsL2bGlK/RJVuXiorsjWoE0irNmLOWFz7ewBXjhzJmYJeowxFp0ZQIpNX5ZNMubpq5mGOGdOM/jxscdTgiLZ4SgbQqhSXl/OSRD+nQJp07zzlMdxUVaQD1EUir4e5c9dQC1mwvZvrkceS0z4w6JJGEoBqBtBr3v72Slxdt4uoJw9QvILIPlAikVZizajv/+/InTDi4J5OPGRR1OCIJRYlAEt6mghJ+8siH9O3chjvO1u8FRPaV+ggkoZWUV/KjR+axp6yCxyaPpUNWetQhiSQcJQJJWO7ODc8v4uO1O/l/3z1S9xESaSQ1DUnCenj2ap6Yu46ffnMwEw7pGXU4IglLiUAS0tvL8vmfF5Zw/EHd+e8ThkYdjkhCUyKQhLN8axH/+eg8Bue0465Jh+tHYyL7SYlAEsrO4jIu+dtc0lNT+Mv3c2mXqW4ukf2lb5EkjNKKSn78yDzW79jDo5PH0rdL26hDEmkVlAgkIbg7Vz+1gNkrtnPnOaMYPUC/HBZpKnFtGjKzCWb2qZnlmdk1tTx/hZktMbMFZva6mfWPZzySuH736mc899EGfnbiUL5zRJ+owxFpVeKWCMwsFbgHOAkYAUwysxE1is0Hct19JPAUcEe84pHENf2DNdz9Rh7nje7Lf/2Hbist0tTiWSMYA+S5+wp3LwNmABNjC7j7G+5eHE7OBnSqJ1/y90Wb+MWzCzluWA63nn6Ibh8hEgfxTAS9gbUx0+vCeXtzMfByHOORBPPe8m1cOmM+o/p24s8XHEFaqi5yE4mHFtFZbGbfBXKBb+zl+SnAFIB+/fo1Y2QSlUXrC5j80FwGdG3LXy8aTduMFrGrirRK8TzFWg/0jZnuE877EjM7AfgFcJq7l9b2Ru4+zd1z3T03JycnLsFKy7FscyEXPvABHduk89APx9KprQaeF4mneCaCOcAQMxtoZhnAecDM2AJmdjhwL0ES2BLHWCRBrNhaxPl/eZ+0FOORS8bSs2NW1CGJtHpxSwTuXgFMBV4BlgJPuPtiM7vZzE4Li/0GaAc8aWYfmdnMvbydJIE124o5/773qapyHps8loHdsqMOSSQpxLXh1d1nAbNqzLsh5v8T4rl8SRxrthUz6b7ZlFRUMn3yOAZ31y2lRZqLeuAkciu2FnH+fe9TUlHJIxePZXivDlGHJJJUlAgkUnlbCpkUNgfNmDKOg3oqCYg0N12YLZFZtL6Ac++dDaAkIBIhJQKJxLvL8zlv2myy0lN5fMo4hmiYSZHIqGlImt3fF23i0unzGdCtLQ/9UJeIikRNiUCa1V/fWcktLy7hsL6deOCi0fqxmEgLoEQgzaKyyrnlxSU8+O4qxo/owV3nHabbRoi0EPomStwVlVZw+Yz5vLZ0C5ccPZBrTx5OqsYZFmkxlAgkrlbm72bKQ3NZkb+bWyYezPe+NiDqkESkBiUCiZs3PtnCpTPmk5ZiPPzDMXx9cLeoQxKRWigRSJOrrHLuen0Zf/rnMob37MC93ztSA82LtGBKBNKkNu8q4bIZ85m9YjtnHtGHW08/hDYZqVGHJSJ1UCKQJvPPTzZz1VML2F1ayW/PHsVZR2rkUZFEoEQg+213aQW3vrSU6R+s4aCe7Zk++XD9UlgkgSgRyH75YOV2rnrqY1ZvL+ZHxw7iihOHkpmmpiCRRKJEII2yq6ScX7/8CY+9v4Y+ndswY/I4xg7qGnVYItIISgSyT9ydWQs3cfOLi9laWMolRw/kihOH6lfCIglM315psM82F3Lj84t5b8U2hvfqwLTv5TKqb6eowxKR/aREIPXKLyrlj68v49H319AuM41bJh7MpDH9SEvVXcxFWgMlAtmr3aUV3PfWCu57cwUlFVWcN7ovV544jC7ZumOoSGuiRCBfUVhSzkPvreYvb61gR3E5Jx/ak5+dOIxBOe2iDk1E4kCJQD63raiUh2ev5q/vrKJgTzn/MSyHy04YymHqBxBp1ZQIhLwtRTzwzkqenreO0ooqThjenZ9+c4g6gkWShBJBkiqrqOLVJZt4ZPZqZq/YTkZaCmce0ZuLjx7I4O76VbBIMlEiSCLuzuINu3j6w3XM/GgD23aX0adzG66aMIxzcvvSrV1m1CGKSASUCJLAss2FvLRwI7MWbuSzzUVkpKZwwojunH1kX44dmqPRwkSSnBJBK1RRWcX8tTt5fekWXl+6mWVbijCD0QO6cMvph3DqyF4aNF5EPqdE0Aq4O2u2F/N2Xj7v5OXzTt42CvaUk5ZijBnYhe+O689Jh/Ske4esqEMVkRZIiSABlVVU8cmmXcxfs5M5q7YzZ9V2Nu8qBaBXxyzGj+jBccNyOHZoDh2y0iOOVkRaOiWCFq6wpJzPNhexdOMulmzcxeINu1i6YRdllVVAcOAfO7Arowd24esHdmVQt2zM1OYvIg2nRNAClJRXsmHnHlZvL2Z1/m5WbStm+dYi8rYUsbGg5PNyHbLSGN6rAz84agAj+3RiVN+O9O7URgd+EdkvcU0EZjYBuAtIBf7i7r+u8Xwm8BBwJLANONfdV8UzpuZUXlnFjt1lbNtdRn5RKVsLS9lSWMrmXSVsKihhY0EJG3buYUth6Zde1y4zjYHdsvnaoK4M7tGOId3bM7xXex30RSQu4pYIzCwVuAcYD6wD5pjZTHdfElPsYmCHuw82s/OA24Fz4xVTXdydiiqnrKKK0ooqSsorKa2oYk9ZJSUVlZSUVVJcVklxeSV7yiooKq2kqKSC3WUVFJaUs2tPBbtKyinYU87O4nJ2Fpexq6Si1mW1zUilV8csDujUhm8MzaFvl7b06dyGvl3aMqBrNt3aZeiALyLNJp41gjFAnruvADCzGcBEIDYRTARuCv9/CrjbzMzdvamDeXzOGqa9uYKKKqei0imvrKKiyimvqKKsMng0ZqlZ6Sm0z0qnQ1Ya7bPS6ZKdwaBu2XRqm0Gntul0bZdJ1+wMumZn0L1DFt3bZ5KdqRY5EWk54nlE6g2sjZleB4zdWxl3rzCzAqArkB9byMymAFMA+vXr16hgumRnclCvDqSlGGkpKaSnGumpKeHDyEhLITMtJfybSmZaCpnpKbRJTyUrfLTNSKVtRhptM1LJzkwjOyNV9+QXkYSXEKem7j4NmAaQm5vbqNrC+BE9GD+iR5PGJSLSGsTzdHY90Ddmuk84r9YyZpYGdCToNBYRkWYSz0QwBxhiZgPNLAM4D5hZo8xM4Pvh/2cB/4xH/4CIiOxd3JqGwjb/qcArBJePPuDui83sZmCuu88E7gceNrM8YDtBshARkWYU1z4Cd58FzKox74aY/0uAs+MZg4iI1E2XvIiIJDklAhGRJKdEICKS5JQIRESSnCXa1ZpmthVY3ciXd6PGr5YTmNal5Wkt6wFal5Zqf9alv7vn1PZEwiWC/WFmc909N+o4moLWpeVpLesBWpeWKl7roqYhEZEkp0QgIpLkki0RTIs6gCakdWl5Wst6gNalpYrLuiRVH4GIiHxVstUIRESkBiUCEZEkl3SJwMxuMbMFZvaRmb1qZgdEHVNjmdlvzOyTcH2eNbNOUcfUGGZ2tpktNrMqM0vIy/zMbIKZfWpmeWZ2TdTxNJaZPWBmW8xsUdSx7A8z62tmb5jZknDfuizqmBrLzLLM7AMz+zhcl/9p8mUkWx+BmXVw913h/5cCI9z9xxGH1ShmdiLBGA4VZnY7gLtfHXFY+8zMhgNVwL3Az9x9bsQh7RMzSwU+A8YTDMk6B5jk7kvqfGELZGbHAkXAQ+5+SNTxNJaZ9QJ6ufuHZtYemAecnqDbxIBsdy8ys3TgbeAyd5/dVMtIuhpBdRIIZQMJmwnd/VV3rwgnZxOMApdw3H2pu38adRz7YQyQ5+4r3L0MmAFMjDimRnH3NwnGBklo7r7R3T8M/y8ElhKMkZ5wPFAUTqaHjyY9biVdIgAws9vMbC1wAXBDfeUTxA+Bl6MOIkn1BtbGTK8jQQ86rZGZDQAOB96POJRGM7NUM/sI2AL8w92bdF1aZSIws9fMbFEtj4kA7v4Ld+8LPApMjTbautW3LmGZXwAVBOvTIjVkPUSampm1A54GLq/RGpBQ3L3S3Q8jqPWPMbMmbbaL6whlUXH3ExpY9FGCEdRujGM4+6W+dTGzi4BTgONb8njP+7BNEtF6oG/MdJ9wnkQobE9/GnjU3Z+JOp6m4O47zewNYALQZB36rbJGUBczGxIzORH4JKpY9peZTQCuAk5z9+Ko40lic4AhZjbQzDIIxt6eGXFMSS3sYL0fWOrud0Ydz/4ws5zqKwLNrA3BRQlNetxKxquGngaGEVylshr4sbsn5NmbmeUBmcC2cNbsRLwCyszOAP4E5AA7gY/c/VuRBrWPzOxk4A9AKvCAu98WbUSNY2bTgeMIbne8GbjR3e+PNKhGMLOjgbeAhQTfdYDrwnHUE4qZjQT+RrBvpQBPuPvNTbqMZEsEIiLyZUnXNCQiIl+mRCAikuSUCEREkpwSgYhIklMiEBFJckoEIiJJTolARCTJ/X/LtFT3h7x0LwAAAABJRU5ErkJggg==\n", + "text/plain": [ + "<Figure size 432x288 with 1 Axes>" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "# Für Plots:\n", + "def plotX(x, X, name, params):\n", + " # Zuerst die Dichtefunktion:\n", + " plt.plot(x, X.pdf(x))\n", + " plt.ylabel('Dichte')\n", + " plt.title(\"PDF von \"+name+\" mit Parametern \"+repr(params))\n", + " plt.show()\n", + " # Dann die kumulative Verteilungsfunktion:\n", + " plt.plot(x, X.cdf(x))\n", + " plt.ylabel('Dichte')\n", + " plt.title(\"CDF von \"+name+\" mit Parametern \"+repr(params))\n", + " plt.show()\n", + "\n", + "x = np.linspace(-3,3,100)\n", + "mu, sigma = 0, 1\n", + "X = st.norm(loc=mu, scale=sigma)\n", + "name = \"Normalverteilung\"\n", + "plotX(x, X, name, (mu, sigma))" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "id": "4a044db8-d054-4673-a359-4f494274d1ce", + "metadata": { + "tags": [] + }, + "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": [ + "X = st.norm(loc=3, scale=2)\n", + "Xsamples = X.rvs(size=10000)\n", + "plt.hist(Xsamples, bins=200, density=True)\n", + "plt.xlabel(\"10000 Samples von N(3,2), in 200 Bins\")\n", + "plt.ylabel(\"Dichte\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "f32228ff-f857-4f93-9774-3739af73a8a0", + "metadata": {}, + "source": [ + "## Kontaminierte Normalverteilung\n", + "\n", + "Die kontaminierte Normalverteilung ist ein einfacher Fall eines sogenannten *gemischten Modells*, genauer einer *Gaussschen Mischung*.\n", + "\n", + "* Parameter $\\varepsilon,\\mu_1,\\sigma_1,\\mu_2,\\sigma_2$, Notation $\\varepsilon\\mathcal{N}(\\mu_1,\\sigma_1^2) + (1-\\varepsilon)\\mathcal{N}(\\mu_2,\\sigma_2^2)$\n", + "* Dichte $\\phi(x) = \\frac{\\varepsilon}{\\sigma\\sqrt{2\\pi}} e^{-\\frac{(x-\\mu_1)^2}{2\\sigma_1^2}} + \\frac{1 - \\varepsilon}{\\sigma\\sqrt{2\\pi}} e^{-\\frac{(x-\\mu_2)^2}{2\\sigma_2^2}}$\n", + "* Erwartungswert $\\varepsilon\\mu_1 + (1-\\varepsilon)\\mu_2$\n", + "* Varianz $\\varepsilon\\sigma_1^2 + (1 - \\varepsilon)\\sigma_2^2 + \\varepsilon(1 - \\varepsilon)(\\mu_1-\\mu_2)^2$\n", + "* Anwendungsbeispiele: Ein zweistufiger Prozess, je nach Ausgang eines Bernoulli-Experiments wird eine andere Normalverteilung gesampelt. Ein konkreter Versuchsaufbau wäre etwa, in einer Informatikvorlesung die Körpergröße der Teilnehmer\\*innen zu messen. Das Geschlecht ist (näherungsweise!) Bernoulli-verteilt (voraussichtlich nicht gleichverteilt), innerhalb der Geschlechtergruppen ist die Körpergröße dann normalverteilt (aber mit verschiedenen Mittelwerten). An diesem Beispiel sieht man auch, dass jede Modellierung das Potential birgt, einen Teil der Realität unsichtbar zu machen, den man möglicherweise gar nicht unsichtbar machen möchte. Zum Glück kann man gemischte Modelle auch mit mehr als einem Parameter $\\varepsilon$ aufstellen." + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "id": "9753d0fd-a039-4d72-8fd1-f80c58ba4c56", + "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": [ + "name = \"0.6*N(0,1) + 0.4*N(2,0.5)\"\n", + "def f(x, epsilon=0.6, mu1=0, sigma1=1, mu2=2, sigma2=0.5):\n", + " norm1 = st.norm(loc=mu1, scale=sigma1)\n", + " norm2 = st.norm(loc=mu2, scale=sigma2)\n", + " return epsilon*norm1.pdf(x) + (1-epsilon)*norm2.pdf(x)\n", + "\n", + "x = np.linspace(-3,3,100)\n", + "plt.plot(x, f(x))\n", + "plt.ylabel('Dichte')\n", + "plt.title(\"PDF von \"+name)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "941767a6-1f68-4bc8-8c43-b6acf078eef3", + "metadata": {}, + "source": [ + "## Dirac-Verteilung\n", + "\n", + "* Parameter $\\xi$, Notation $\\delta(\\xi)$\n", + "* Ohne Dichte, aber $P(X = \\xi) = 1$\n", + "* Erwartungswert $\\xi$\n", + "* Varianz $0$\n", + "* Entropie $0$\n", + "* Anwendungsbeispiele:\n", + " * 'Dichtefunktionen' im stetigen Sinne für diskrete Verteilungen hinschreiben\n", + " * Fotos vom Sternenhimmel (z.B. mit Hubble) - jeder Stern ist ein verschmierter Pixelhaufen (normalverteilt), aber die 'eigentliche' Verteilung der Sterne dahinter ist eine Summe von Dirac-Maßen, denn jeder Stern ist an genau einem Punkt (die tatsächliche Ausdehnung des Sterns ist aufgrund der Entfernung irrelevant). Mit dieser Modellierung arbeiten manche Superresolution-Techniken.\n", + " \n", + "Die Dirac-Verteilung ist strenggenommen unplotbar (denn es gibt ja keine Dichte). Manche zeichnen zur Illustration einen Pfeil der Länge $1$ bei $\\xi$ ein.\n", + "Die kumulative Verteilungsfunktion ist eine Heaviside-Step-Funktion:" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "id": "0da093d7-7476-4088-9ea8-ae9192e436fb", + "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": [ + "name = \"Dirac-Verteilung bei ξ=0\"\n", + "plt.plot(x, np.heaviside(x, 1))\n", + "plt.ylabel('Dichte')\n", + "plt.title(\"CDF von \"+name)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "5f9e0ec5-2781-4e56-8258-9979887a30bd", + "metadata": {}, + "source": [ + "## Stetige Gleichverteilung\n", + "\n", + "* Parameter $a,b$, Notation $Uni(a,b)$\n", + "* Dichte $\\frac{1}{b-a}$\n", + "* Erwartungswert $\\frac{a+b}{2}$\n", + "* Varianz $\\frac{(a-b)^2}{12}$\n", + "* Maximale Entropie unter allen Verteilungen mit Träger $[a,b]$\n", + "* Anwendungsbeispiele:\n", + " * Wenn ein Bus an Ihrer Haltestelle extrem pünktlich jede Stunde kommt, Sie aber nicht wissen, wann er das letzte Mal gefahren ist, so ist ihre Wartezeit an der Haltestelle in Minuten gleichverteilt in $[0,60]$.\n", + " * Die Temperatur ihrer CPU ab der dritten Nachkommastelle `(T * 100 - int(T * 100))` ist annähernd gleichverteilt." + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "id": "94ab9253-19e3-4256-94b6-0132cdf2790f", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "0.5 0.75\n" + ] + }, + { + "data": { + "image/png": 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\n", + "text/plain": [ + "<Figure size 432x288 with 1 Axes>" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + }, + { + "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": [ + "a, b = -1, 2\n", + "X = st.uniform(loc=a, scale=b-a)\n", + "print(X.mean(), X.var())\n", + "name = \"Gleichverteilung\"\n", + "plotX(x, X, name, (a, b))" + ] + }, + { + "cell_type": "markdown", + "id": "e48381a4-8aea-42a5-ace1-ca83ab4e6c14", + "metadata": {}, + "source": [ + "## Exponentialverteilung\n", + "\n", + "* Parameter $\\lambda$, Notation $Exp(\\lambda)$\n", + "* Dichte $\\lambda e^{-\\lambda x}$\n", + "* Erwartungswert $\\dfrac{1}{\\lambda}$\n", + "* Varianz $\\dfrac{1}{\\lambda^2}$\n", + "* Maximale Entropie unter allen Verteilungen mit Träger $[0,\\infty)$ und Erwartungswert $\\dfrac{1}{\\lambda}$\n", + "* Anwendungsbeispiele:\n", + " * Wie lange dauert es, bis auf dem Server das nächste Paket ankommt?" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "id": "dc79a2c7-07ac-4dc7-8bed-859ad3a629bf", + "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" + }, + { + "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": [ + "name = \"Exponentialverteilung\"\n", + "lamda = 2\n", + "X = st.expon(scale = 1/lamda)\n", + "x = np.linspace(-1,5,100)\n", + "plotX(x,X,name,(lamda))" + ] + }, + { + "cell_type": "markdown", + "id": "2eadac49-cfe0-4715-9f65-743a969eefad", + "metadata": {}, + "source": [ + "## Lognormalverteilung\n", + "\n", + "* Parameter $\\mu, \\sigma$, Notation $Lognormal(\\mu,\\sigma^2)$\n", + "* Dichte $\\frac{1}{x\\sigma\\sqrt{2\\pi}} e^{-\\frac{(log(x)-\\mu)^2}{2\\sigma^2}}$\n", + "* Erwartungswert $e^{\\mu + \\frac{\\sigma^2}{2}}$ aber $\\mathbb{E}(log X) = \\mu$ \n", + "* Varianz $(e^{\\sigma^2}-1)e^{2\\mu + \\sigma^2}$ aber $\\mathbb{V}(log X) = \\sigma^2$ \n", + "* Maximale Entropie unter allen Verteilungen mit Träger $(0,\\infty)$ und $\\mathbb{E}(\\log X) = \\mu$ und $\\mathbb{V}(\\log X) = \\sigma^2$\n", + "* Anwendungsbeispiele:\n", + " * Die Länge eines Schachspiels ist log-normalverteilt\n", + " * Viele (!) weitere Beispiele hat die [Wikipedia zur Lognormalverteilung](https://en.wikipedia.org/wiki/Log-normal_distribution#Occurrence_and_applications)\n", + " " + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "id": "c6db802c-3fd7-4d8b-a4fb-fe13950dfc46", + "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" + }, + { + "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": [ + "name = \"Log-normalverteilung\"\n", + "mu, sigma = 1, 2\n", + "X = st.lognorm(s=sigma, scale = np.exp(mu))\n", + "x = np.linspace(0,2,1000)\n", + "plotX(x,X,name,(mu, sigma))" + ] + }, + { + "cell_type": "markdown", + "id": "7c3b2bd4-ded0-4f92-bbce-7870a067a626", + "metadata": {}, + "source": [ + "## Weibull-Verteilung\n", + "\n", + "* Parameter $\\lambda>0, k>0$, Notation $Weibull(\\lambda,k)$\n", + "* Dichte $f(x) = \\dfrac{k}{\\lambda} {\\left(\\frac{x}{\\lambda}\\right)}^{k-1} e^{-{\\left(\\frac{x}{\\lambda}\\right)}^k}$ für $x \\geq 0$ \\\n", + " und $f(x)=0$ für $x < 0$\n", + "* Erwartungswert $\\lambda \\Gamma(1+\\frac{1}{k})$ (wobei $\\Gamma(n+1) = n!$)\n", + "* Maximale Entropie unter allen Verteilungen mit einer Bedingung an alle Momente: $\\mathbb{E}(X^n) = \\lambda^n$ und einer Bedingung an $\\mathbb{E}(\\log X)$.\n", + "* Anwendungsbeispiele:\n", + " * Wie lange läuft eine Wärmepumpe, bis sie kaputt geht?\n", + " * Wie hoch ist die Windgeschwindigkeit?\n", + " * Was ist die höchste Niederschlagsmenge an einem einzelnen Tag im Jahr?\n", + " " + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "id": "859f3280-1451-48d6-8632-ab225a7d09cf", + "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" + }, + { + "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": [ + "name = \"Weibull-verteilung\"\n", + "lamda, k = 1, 1.3\n", + "X = st.weibull_min(c=k, loc=0, scale = lamda)\n", + "x = np.linspace(0,2,1000)\n", + "plotX(x,X,name,(lamda, k))" + ] + }, + { + "cell_type": "markdown", + "id": "924ecdb7-fcb3-4cd9-b93a-687780a6edaf", + "metadata": {}, + "source": [ + "# Daten zum plotten und fitten\n", + "\n", + "Damit wir ein bisschen üben können, Daten zu plotten und Verteilungen auf Daten zu fitten (und das dann wiederum im Plot zu visualisieren), brauchen wir Spielzeugdaten.\n", + "\n", + "Ein sehr einfacher und daher sehr instruktiver Datensatz ist Fisher-Anderson's 'iris flower dataset', den wir z.B. über Scikit-Learn beziehen können:" + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "id": "c6fc6939-434a-4cd7-957c-ed09626cc5e9", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[[5.1 3.5 1.4 0.2]\n", + " [4.9 3. 1.4 0.2]\n", + " [4.7 3.2 1.3 0.2]\n", + " [4.6 3.1 1.5 0.2]\n", + " [5. 3.6 1.4 0.2]]\n" + ] + } + ], + "source": [ + "from sklearn import datasets\n", + "iris = datasets.load_iris()\n", + "print(iris[\"data\"][:5])" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "id": "a01fd47d-0f25-47f9-9c3d-328be854fcfa", + "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": [ + "first_feature = iris[\"data\"][:,0]\n", + "plt.hist(first_feature, bins=30, density=True)\n", + "# Jetzt die best-fitting-Normalverteilung rein:\n", + "mu, sigma = st.norm.fit(first_feature)\n", + "X = st.norm(loc=mu, scale=sigma)\n", + "x = np.linspace(3,9,100)\n", + "plt.plot(x, X.pdf(x))\n", + "plt.xlabel(\"Erstes Feature Iris-Datensatz\")\n", + "plt.ylabel(\"Dichte\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "25e1bab3-afe7-4bb4-a1fd-bc05f1e75abe", + "metadata": {}, + "source": [ + "Eine Quelle für relativ gut gepflegte Spielzeugdaten, die auch viel diskutiert werden, ist [Kaggle](https://www.kaggle.com/datasets).\n", + "Um diese Daten zu verarbeiten, muss man meist CSV-Dateien in Numpy-Arrays verwandeln. Besonders entspannt geht das mit Pandas (das auf Numpy aufbaut), womit wir uns noch beschäftigen werden. Für den Anfang reicht auch Numpy's eingebautes `np.genfromtxt`.\n", + "\n", + "Der deutsche Wetterdienst DWD hat mit dem [Climate Data Center](https://cdc.dwd.de/portal/) eine große Sammlung gut dokumentierter, teilweise täglich aktualisierter Wetter- und Klimadaten. Darin sind Variablen mit räumlicher Verteilung und Zeitreihen verschiedenster physikalischer Größen und statistischer Verteilungen zu finden. Interessant (und beantwortbar) ist z.B. die Frage, ob Starkregenereignisse in Düsseldorf in den letzten Jahrzehnten zugenommen haben.\n", + "\n", + "Schauen Sie sich diese (und andere!) Daten an, finden Sie heraus, wie die Variablen verteilt sein könnten, plotten und fitten Sie Verteilungen. Besonders durch den Vergleich des Histogramms mit der Dichtefunktion der gefitteten Verteilung sehen Sie, wie nah wir liegen.\n", + "\n", + "Da sehr viele Daten nicht einer der hier aufgezählten Verteilungen genügen, benötigen wir noch komplexere Modelle. Diese sind allerdings zumeist mehrstufig aus den einfachen Modellen zusammengesetzt, wie bei der kontaminierten Normalverteilung. Eines der wichtigsten Modelle wird später die Mischung multivariater Normalverteilungen sein, anhand dessen wir einige Prinzipien des maschinellen Lernens betrachten können.\n", + "\n", + "Wenn Sie sich weiter mit den Zusammenhängen zwischen den Verteilungen beschäftigen wollen, und dabei nicht übersehen haben, dass sich einige Verteilungen durchaus verschieden parametrisieren lassen (z.B. die Normalverteilung mit $\\mu,\\sigma$ aber auch mit $\\mu,\\sigma^2$), könnte ProbOnto das Richtige sein. Es handelt sich dabei um eine Ontologie und Wissensbasis, die ursprünglich für die Pharmakologie entwickelt wurde. Die Übersichtsgrafik ist bereits eine gute Gedächtnisstütze:\n", + "\n", + "\n", + "[ProbOnto 2.5](https://commons.wikimedia.org/wiki/File:ProbOnto2.5.jpg) CC-BY-SA [Wikipedia user Fuzzyrandom](https://commons.wikimedia.org/w/index.php?title=User:Fuzzyrandom)" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.8.10" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +}