{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Δημιουργία τυχαίου δείγματος 1024 στοιχείων από την $\\mathcal N (2,5^2)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "dataset = pd.DataFrame(5*np.random.randn(1024)+2, columns=['x'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>x</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>-2.488147</td>\n",
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       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>6.570618</td>\n",
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       "      <th>2</th>\n",
       "      <td>5.296783</td>\n",
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       "      <th>3</th>\n",
       "      <td>-0.793945</td>\n",
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       "      <th>4</th>\n",
       "      <td>-3.796571</td>\n",
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      "text/plain": [
       "          x\n",
       "0 -2.488147\n",
       "1  6.570618\n",
       "2  5.296783\n",
       "3 -0.793945\n",
       "4 -3.796571"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dataset.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Υπολογισμός του συντελεστή ασυμμετρίας Fisher-Pearson"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$G_1 = \\frac{N^2}{(N-1)(N-2)}\\frac{\\frac{1}{N}\\sum_{n=1}^N(x_n-\\bar{X})^3}{s^3} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x    0.023114\n",
       "dtype: float64"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dataset.skew()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Με τη len() λαμβάνουμε πόσα στοιχεία έχει το σύνολο δεδομένων"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = len(dataset)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Υπολογισμός του $\\sum_{n=1}^N(x_n-\\bar{X})^3$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x    3144.619736\n",
       "dtype: float64"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "((dataset - dataset.mean())**3).sum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "((dataset - dataset.mean())**3).sum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "sum3 = ((dataset - dataset.mean())**3).sum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "s = dataset.std()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "G1 = N**2 / ((N-1)*(N-2)) * (1/N) * sum3 / s**3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x    0.023114\n",
       "dtype: float64"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Υπογογισμός του μέτρου κύρτωσης"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\mathrm{kurt} = \\frac{\\frac{1}{N}\\sum_{n=1}^N(x_n-\\bar{X})^4}{s^4}-3$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x    0.149248\n",
       "dtype: float64"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dataset.kurt()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Άσκηση : Υπολογισμός του $\\sum_{n=1}^N(x_n-\\bar{X})^4$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "sum4 = ((dataset-dataset.mean())**4).sum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "sum4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "kurt = (1/N * sum4)/s**4 - 3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "kurt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Άσκηση : Υπολογίστε τη μέση τιμή, τη διάμεσο, τα τεταρτημόρια για το dataset"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dataset.describe()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Άσκηση : Δημιουργήστε το Box-and-Whisker plot για το dataset"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dataset.plot.box()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## scipy.stats"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "import scipy.stats as st"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### z-scores"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$z = \\frac{x-\\bar{X}}{s}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "x1 = 1.0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "z1 = (x1 - dataset.mean())/dataset.std()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x   -0.188914\n",
       "dtype: float64"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Υπολογισμός του $P(Z \\leq z_1)$ με την scipy.stats"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "P1 = st.norm.cdf(z1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.42507998])"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.7217573815286908"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z1 = -8/9\n",
    "z2 = 4/3\n",
    "P1 = st.norm.cdf(z1)\n",
    "P2 = st.norm.cdf(z2)\n",
    "P2-P1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "z1 = -1.602"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "P1 = st.norm.cdf(z1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.054577804746102826"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Άσκηση : Επιλύστε με χρήση των pandas και scipy.stats\n",
    "Σε ένα δημοψήφισμα με απάντηση ΝΑΙ / ΟΧΙ, το ΟΧΙ συγκέντρωσε το 60% των συνολικών ψήφων. Ποιά η πιθανότητα σε ένα τύχαιο δείγμα 500 ψηφοφόρων το ΝΑΙ να συγκεντρώνει τουλάχιστον το 50%;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "p = 0.6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "sigma_p_hat = (0.6*(1-0.6)/500)**(1/2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "z = (0.5-p)/sigma_p_hat"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "st.norm.cdf(z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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