{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Projekt - Test t studenta\n",
    "\n",
    "- Marcin Kostrzewski\n",
    "- Krystian Wasilewski\n",
    "- Mateusz Tylka\n",
    "\n",
    "## Test t studenta\n",
    "\n",
    "Metoda statystyczna służącą do porównania dwóch średnich między sobą gdy znamy liczbę badanych próbek, średnią arytmetyczną oraz wartość odchylenia standardowego lub wariancji.\n",
    "Jest to jeden z mniej skomplikowanych i bardzo często wykorzystywanych testów statystycznych używanych do weryfikacji hipotez. Dzięki niemu możemy dowiedzieć się czy dwie różne średnie są różne niechcący (w wyniku przypadku) czy są różne istotnie statystycznie (np. z uwagi na naszą manipulację eksperymentalna).\n",
    "Wyróżniamy 3 wersję testu t:\n",
    "\n",
    "1. test t Studenta dla jednej próby\n",
    "2. test t Studenta dla prób niezależnych  \n",
    "3. test t Studenta dla prób zależnych"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Test Shapiro Wilka\n",
    "\n",
    "Wszystkie rodzaje testów są testami parametrycznymi, a co za tym idzie nasze mierzone zmienne ilościowe powinny mieć rozkład normalny.  \n",
    "Dzięki testowi Shapiro Wilka możemy sprawdzić to założenie."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "## Testowanie hipotez metodą bootstrap\n",
    "\n",
    "**Bootstrap** – metoda szacowania (estymacji) wyników poprzez wielokrotne losowanie ze zwracaniem z próby. Polega ona na utworzeniu nowego rozkładu wyników, na podstawie posiadanych danych, poprzez wielokrotne losowanie wartości z posiadanej próby. Metoda ze zwracaniem polega na tym, że po wylosowaniu danej wartości, “wraca” ona z powrotem do zbioru.\n",
    "\n",
    "Metoda bootstrapowa znajduje zastosowanie w sytuacji, w której nie znamy rozkładu z populacji z której pochodzi próbka lub w przypadku rozkładów małych lub asymetrycznych. W takim wypadku, dzięki tej metodzie, wyniki testów parametrycznych i analiz opartych o modele liniowe są bardziej precyzyjne. Zazwyczaj losuje się wiele próbek, np. 2000 czy 5000."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "# Definicje funkcji"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "from enum import Enum\n",
    "from scipy.stats import ttest_ind, ttest_1samp, ttest_rel, shapiro"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [],
   "source": [
    "dataset = pd.read_csv('experiment_data.csv') # TODO: del?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [],
   "source": [
    "class Alternatives(Enum):\n",
    "    LESS = 'less'\n",
    "    GREATER = 'greater'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [],
   "source": [
    "def calculate_t_difference(t_stat_sample, t_stat_list, alternative):\n",
    "    \"\"\"\n",
    "    Funkcja oblicza procent statystyk testowych powstałych z prób bootstrapowych, \n",
    "    które róznią się od statystyki testowej powstałej ze zbioru według hipotezy alternatywnej.\n",
    "    \"\"\"\n",
    "    all_stats = len(t_stat_list)\n",
    "    stats_different_count = 0\n",
    "    for t_stat_boot in t_stat_list:\n",
    "        if alternative is Alternatives.LESS and t_stat_boot > t_stat_sample:\n",
    "            stats_different_count += 1 \n",
    "        elif alternative is Alternatives.GREATER and t_stat_boot < t_stat_sample:\n",
    "            stats_different_count += 1\n",
    "    return stats_different_count / all_stats"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "def t_test_1_samp(sample_1, population_mean=None, alternative=Alternatives.LESS):\n",
    "    \"\"\"\n",
    "    Funkcja przeprowadza test T-studenta dla jednej zmiennej.\n",
    "    \"\"\"\n",
    "    t_stat_from_sample, _ = ttest_1samp(a=sample_1, popmean=population_mean, alternative=alternative.value)\n",
    "    t_stat_list = get_t_stats(sample_1, t_stat_fn=ttest_1samp, alternative=alternative, population_mean=population_mean)\n",
    "\n",
    "    p = calculate_t_difference(t_stat_from_sample, t_stat_list, alternative)\n",
    "\n",
    "    return p, t_stat_from_sample, t_stat_list"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [],
   "source": [
    "def t_test_ind(sample_1, sample_2, alternative=Alternatives.LESS):\n",
    "    \"\"\"\n",
    "    Funkcja przeprowadza test T-studenta dla dwóch zmiennych niezależnych.\n",
    "    \"\"\"\n",
    "    t_stat_from_sample, _ = ttest_ind(sample_1, sample_2, alternative=alternative.value)\n",
    "    t_stat_list = get_t_stats(sample_1, sample_2, alternative=alternative, t_stat_fn=ttest_ind)\n",
    "\n",
    "    p = calculate_t_difference(t_stat_from_sample, t_stat_list, alternative)\n",
    "\n",
    "    return p, t_stat_from_sample, t_stat_list"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [],
   "source": [
    "def t_test_dep(sample_1, sample_2, alternative=Alternatives.LESS):\n",
    "    \"\"\"\n",
    "    Funkcja przeprowadza test T-studenta dla dwóch zmiennych zależnych.\n",
    "    \"\"\"\n",
    "    t_stat_list = get_t_stats(sample_1, sample_2, alternative=alternative, t_stat_fn=ttest_rel)\n",
    "    t_stat_from_sample, _ = ttest_rel(sample_1, sample_2, alternative=alternative.value)\n",
    "\n",
    "    p = calculate_t_difference(t_stat_from_sample, t_stat_list, alternative)\n",
    "\n",
    "    return p, t_stat_from_sample, t_stat_list"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_t_stats(sample_1, sample_2=None, t_stat_fn=ttest_1samp, alternative=Alternatives.LESS, population_mean=None):\n",
    "    \"\"\"Funkcja oblicza listę statystyk testowych dla każdej próbki bootstrapowej wybranej na podstawie danych sample_1 i sample_2\"\"\"\n",
    "    t_stat_list = []\n",
    "\n",
    "    # One sample test\n",
    "    if t_stat_fn is ttest_1samp and sample_2 is None:\n",
    "        if not population_mean:\n",
    "            raise Exception(\"population_mean not provided\")\n",
    "        for bootstrap in generate_bootstraps(sample_1):\n",
    "            stat, _ = t_stat_fn(bootstrap, population_mean, alternative=alternative.value)\n",
    "            t_stat_list.append(stat)\n",
    "        return t_stat_list\n",
    "\n",
    "    # Two sample test\n",
    "    for bootstrap_sample in generate_bootstraps(pd.concat((sample_1, sample_2), ignore_index=True)):\n",
    "        bootstrap_1 = bootstrap_sample.iloc[: len(bootstrap_sample) // 2]\n",
    "        bootstrap_2 = bootstrap_sample.iloc[len(bootstrap_sample) // 2 :]\n",
    "        stat, _ = t_stat_fn(bootstrap_1, bootstrap_2, alternative=alternative.value)\n",
    "        t_stat_list.append(stat)\n",
    "    return t_stat_list"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [],
   "source": [
    "def pretty_print_test(p, t_stat_from_sample, t_stat_list, thesis, alternative, max_print=5):\n",
    "    print('Wyniki bootstrapowej wersji testu T-studenta')\n",
    "    print()\n",
    "    print(f'Hipoteza: {thesis}')\n",
    "    if alternative is Alternatives.LESS:\n",
    "        print(f'Hipoteza alternatywna: średnia jest mniejsza')\n",
    "    else:\n",
    "        print(f'Hipoteza alternatywna: średnia jest większa')\n",
    "    print()\n",
    "    print(f'p: {p}')\n",
    "    print(f'Wartość statystyki testowej z próby: {t_stat_from_sample}')\n",
    "    print(f'Wartości statystyk z prób boostrapowych:')\n",
    "\n",
    "    t_stat_list_len = len(t_stat_list)\n",
    "    for i in range(min(max_print, t_stat_list_len)):\n",
    "        print(f'{t_stat_list[i]}, ', end='')\n",
    "    if max_print < t_stat_list_len:\n",
    "        remaining = t_stat_list_len - max_print\n",
    "        print(f'... (i {remaining} pozostałych)')\n",
    "\n",
    "    print()\n",
    "    print()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "def generate_bootstraps(data, n_bootstraps=1000):\n",
    "    data_size = data.shape[0]\n",
    "    for _ in range(n_bootstraps):\n",
    "        indices =  np.random.choice(len(data), size=data_size)\n",
    "        yield data.iloc[indices, :]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "def bootstrap_one_sample(sample, population_mean, alternative=Alternatives.LESS):\n",
    "    p, t, ts = t_test_1_samp(\n",
    "        sample_1=sample,\n",
    "        population_mean=population_mean,\n",
    "        alternative=alternative,\n",
    "    )\n",
    "    \n",
    "    pretty_print_test(p, t, ts, f'średnia jest równa {population_mean}', alternative)\n",
    "    print()\n",
    "    return p, t, ts"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "def bootstrap_independent(sample_1, sample_2, alternative=Alternatives.LESS):\n",
    "    p, t, ts = t_test_ind(\n",
    "        sample_1=sample_1,\n",
    "        sample_2=sample_2,\n",
    "        alternative=alternative,\n",
    "    )\n",
    "    \n",
    "    pretty_print_test(p, t, ts, 'średnie są takie same', alternative)\n",
    "    return p, t, ts"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "def bootstrap_dependent(sample_1, sample_2, alternative=Alternatives.LESS):\n",
    "    p, t, ts = t_test_dep(\n",
    "        sample_1=sample_1,\n",
    "        sample_2=sample_2,\n",
    "        alternative=alternative,\n",
    "    )\n",
    "    \n",
    "    pretty_print_test(p, t, ts, 'średnie są takie same', alternative)\n",
    "    return p, t, ts"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "def draw_distribution(stats, comparision_value):\n",
    "    \"\"\"\n",
    "    Funkcja rysuje rozkład statystyki testowej\n",
    "    @param stats: lista statystyk testowych\n",
    "    @param comparision_value: pierwotna próbka\n",
    "    \"\"\"\n",
    "    plt.hist(stats)\n",
    "    plt.axvline(comparision_value, color='red')\n",
    "    plt.xlabel('Test statistic value')\n",
    "    plt.ylabel('Frequency')\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "# Wczytanie danych"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "0    169.5557\ndtype: float64\n0    175.1417\ndtype: float64\n0    79.6342\ndtype: float64\n0    76.5602\ndtype: float64\n"
     ]
    }
   ],
   "source": [
    "dataset = pd.read_csv('experiment_data.csv')\n",
    "heights_female = pd.DataFrame(dataset['Female height'].to_numpy())  # xd\n",
    "heights_male = pd.DataFrame(dataset['Male height'].to_numpy())\n",
    "weights_before = pd.DataFrame(dataset['Weight before'].to_numpy())\n",
    "weights_after = pd.DataFrame(dataset['Weight after'].to_numpy())\n",
    "print(np.mean(heights_female))\n",
    "print(np.mean(heights_male))\n",
    "print(np.mean(weights_before))\n",
    "print(np.mean(weights_after))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "# Jedna próba\n",
    "\n",
    "**Test t Studenta dla jednej próby** wykorzystujemy gdy chcemy porównać średnią “teoretyczną” ze średnią, którą faktycznie możemy zaobserwować w naszej bazie danych. Średnia teoretyczna to średnia pochodząca z innych badań lub po prostu bez większych uzasadnień pochodząca z naszej głowy.\n",
    "\n",
    "Wyobraźmy sobie, że mamy dane z takimi zmiennymi jak wzrost pewnej grupy ludzi. Dzięki testowi t Studenta dla jednej próby możemy dowiedzieć się np. czy wzrost naszego młodszego brata wynoszący 160cm odbiega znacząco od średniej wzrostu tej grupy.\n",
    "\n",
    "### Hipoteza\n",
    "\n",
    "*H0: Badana próba została wylosowana z populacji, w której wzrost osób wynosi średnio 160cm.*  \n",
    "*H1: Badana próba została wylosowana z populacji gdzie średni wzrost jest większy 160cm.*\n",
    "\n",
    "### Sprawdzenie założeń\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "p = 0.791\n"
     ]
    }
   ],
   "source": [
    "# Sprawdzamy, czy próby mają rozkład normalny\n",
    "shapiro_test = shapiro(heights_female)\n",
    "print(f\"p = {round(shapiro_test.pvalue,4)}\")"
   ]
  },
  {
   "source": [
    "P wartość jest większa niż alfa = 0.05, więc próba ma prawdopodobnie rozkład normalny. Możemy stostować testy."
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "## Test\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "Wyniki bootstrapowej wersji testu T-studenta\n\nHipoteza: średnia jest równa 160.0\nHipoteza alternatywna: średnia jest większa\n\np: 0.5\nWartość statystyki testowej z próby: [19.1207964]\nWartości statystyk z prób boostrapowych:\n[17.41702865], [19.17874674], [20.59090525], [17.666445], [19.3593138], ... (i 95 pozostałych)\n\n\n\n"
     ]
    },
    {
     "output_type": "display_data",
     "data": {
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\n"
     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "tested_mean = 160.0\n",
    "\n",
    "p, t, ts = bootstrap_one_sample(heights_female, tested_mean, alternative=Alternatives.GREATER)\n",
    "draw_distribution([x[0] for x in ts], t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "## Wniosek\n",
    "\n",
    "Nie mamy podstaw, żeby odrzucić hipotezę zerową mówiącą, że średnia wynosi 160."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "# Dwie próby niezależne\n",
    "\n",
    "**Test t Studenta dla prób niezależnych** jest najczęściej stosowaną metodą statystyczną w celu porównania średnich z dwóch niezależnych od siebie grup. Wykorzystujemy go gdy chcemy porównać dwie grupy pod względem jakiejś zmiennej ilościowej. Na przykład gdy chcemy porównać średni wzrost kobiet i mężczyzn w danej grupie.\n",
    "Zazwyczaj dwie średnie z różnych od siebie grup będą się różnić. Test t Studenta powie nam jednak czy owe różnice są istotne statystycznie – czy nie są przypadkowe.\n",
    "Jeśli wynik testu t Studenta będzie istotny na poziomie p < 0,05 możemy odrzucić hipotezę zerową na rzecz hipotezy alternatywnej.\n",
    "\n",
    "## Hipoteza\n",
    "\n",
    "*H0: Średni wzrost w grupie mężczyzn jest taki sam jak średni w grupie kobiet. Hipoteza alternatywna z kolei*  \n",
    "*H1: Kobiety będą niższe od mężczyzn pod względem wzrostu.*\n",
    "\n",
    "## Sprawdzenie założeń\n",
    "\n",
    "Założenie o rozkładzie normalnym danych - sprawdzane testem Shapiro-Wilka"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "p = 0.791\np = 0.7535\n"
     ]
    }
   ],
   "source": [
    "shapiro_test = shapiro(heights_female)\n",
    "print(f\"p = {round(shapiro_test.pvalue,4)}\")\n",
    "\n",
    "shapiro_test = shapiro(heights_male)\n",
    "print(f\"p = {round(shapiro_test.pvalue,4)}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "Wartości **p** w teście Shapiro-Wilka powyżej **0.05** -> Dane prawdopodobnie mają rozkład normalny"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "## Test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "Wyniki bootstrapowej wersji testu T-studenta\n\nHipoteza: średnie są takie same\nHipoteza alternatywna: średnia jest mniejsza\n\np: 0.0\nWartość statystyki testowej z próby: [8.04931557]\nWartości statystyk z prób boostrapowych:\n[0.2748409], [-0.61193473], [1.24335163], [-2.56879464], [0.34249038], ... (i 95 pozostałych)\n\n\n"
     ]
    },
    {
     "output_type": "display_data",
     "data": {
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\n"
     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "p, t, ts = bootstrap_independent(heights_male, heights_female)\n",
    "ts = [x[0] for x in ts]\n",
    "draw_distribution(ts, t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "## Wniosek"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "# Dwie próby zależne\n",
    "\n",
    "W odróżnieniu od testu dla prób niezależnych, gdzie porównujemy dwie grupy, ten rodzaj testu stosujemy gdy poddajemy analizie tą samą pojedynczą grupę, ale dwukrotnie w czasie.\n",
    "\n",
    "**Przykład**: Porównane zostały wagi przed dietą i po diecie.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "### Hipoteza\n",
    "H0 - Średnia waga nie uległa zmianie po zastosowaniu diety\n",
    "H1 - Średnia waga po diecie jest znacząco mniejsza od wagi przed dietą\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "\n",
    "### Sprawdzenie założeń\n",
    "\n",
    "Założenie o rozkładzie normalnym danych - sprawdzane testem Shapiro-Wilka"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "p = 0.3308\np = 0.4569\n"
     ]
    }
   ],
   "source": [
    "shapiro_test = shapiro(weights_before)\n",
    "print(f\"p = {round(shapiro_test.pvalue,4)}\")\n",
    "\n",
    "shapiro_test = shapiro(weights_after)\n",
    "print(f\"p = {round(shapiro_test.pvalue,4)}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "Wartości **p** w teście Shapiro-Wilka powyżej **0.05** -> Dane prawdopodobnie mają rozkład normalny"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "## Test"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "Wyniki bootstrapowej wersji testu T-studenta\n\nHipoteza: średnie są takie same\nHipoteza alternatywna: średnia jest mniejsza\n\np: 0.0\nWartość statystyki testowej z próby: [48.30834167]\nWartości statystyk z prób boostrapowych:\n[-0.18332849], [-1.21537352], [1.64628473], [1.06552535], [-0.71420173], ... (i 95 pozostałych)\n\n\n"
     ]
    },
    {
     "output_type": "display_data",
     "data": {
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\n"
     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "p, t, ts = bootstrap_dependent(weights_before, weights_after)\n",
    "ts = [x[0] for x in ts]\n",
    "draw_distribution(ts, t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "## Wniosek\n",
    "\n",
    "???"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "interpreter": {
   "hash": "1b132c2ed43285dcf39f6d01712959169a14a721cf314fe69015adab49bb1fd1"
  },
  "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-final"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}