path: root/sem6/prob/miniexam/Untitled.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy import stats"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Problem 1\n",
    "\n",
    "Man kan sige at distribution af et enkelt kast af den 8 kantede terning er $1/8$ for hver værdi i $[1, 8]$.\n",
    "Ved at summe dem sammen kommer de til at ligne en gaussian random variable.\n",
    "\n",
    "Man ved også de to limits for distributionen, nemlig $[50 \\cdot 1, 50 \\cdot 8]$, og hver imellem må mean ligge.\n",
    "Ud fra intervallet og mean kan man regne varience."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Man kan først finde størrelsen af intervallet, eller var * 2\n",
    "var2 = 50 * 8 - 50 * 1\n",
    "sigma2 = var2/2\n",
    "mean = 50 * 1 + sigma2\n",
    "\n",
    "# Ved coin flip får man egentlig det samme distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Part A\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<BarContainer object of 500 artists>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "slots = np.zeros(500)\n",
    "x = np.arange(500)\n",
    "for exp in range(1000):\n",
    "    s = 0\n",
    "    for i in range(50):\n",
    "        r = np.random.randint(1, 9)\n",
    "        #r = [1, 8][np.random.randint(0, 2)]\n",
    "        s += r\n",
    "    slots[s] += 1\n",
    "\n",
    "plt.bar(x, slots)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Problem 2\n",
    "\n",
    "The distribution function is given by:\n",
    "$$\n",
    "f(x)_\\theta = \\frac x {\\theta^2} e ^ {- \\frac x \\theta }\n",
    "$$\n",
    "\n",
    "Then we can find the likelyhood function $f(x_1,...,x_n | \\theta)$.\n",
    "\n",
    "$$\n",
    "f(x_1,...,x_n | \\theta) = f(x_1)_\\theta \\cdot ... \\cdot f(x_n)_\\theta = \\frac {\\prod_i x_i} {\\theta^{2n}} \\cdot \\exp \\left( -\\frac {\\sum_i} \\theta \\right)\n",
    "$$\n",
    "\n",
    "Her kan man tage log på hver side\n",
    "\n",
    "$$\n",
    "\\log f(x_1,...,x_n | \\theta) = \\log \\left( \\frac {\\prod_i x_i} {\\theta^{2n}} \\right) - \\frac 1 \\theta \\sum_i x_i\n",
    "$$\n",
    "\n",
    "Og diff i forhold til $\\theta$.\n",
    "\n",
    "$$\n",
    "\\frac d {d\\theta} \\log f(x_1,...,x_n | \\theta) = -\\frac {2n} \\theta + \\frac 1 {\\theta^2} \\sum_i x_i\n",
    "$$\n",
    "\n",
    "Og man kan løse for $0$.\n",
    "\n",
    "$$\n",
    "\\theta = \\frac {\\sum_i x_i} {2n}\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Estimated theta = 1.2944444444444445\n"
     ]
    }
   ],
   "source": [
    "# Part B\n",
    "# Her indsættes samples\n",
    "\n",
    "samples = np.array([3.2, 1.4, 6.5, 2.2, 1.8, 2.6, 3.9, 0.5, 1.2])\n",
    "N = len(samples)\n",
    "\n",
    "sum_samples = np.sum(samples)\n",
    "\n",
    "theta_est = sum_samples / (2 * N)\n",
    "print(f\"Estimated theta = {theta_est}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "argmax: 1.2962962962962963\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7faf4d186640>]"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "# Prøver lige at løse den programmerbart også\n",
    "thetas = np.linspace(0, 5, 1000)[1:]\n",
    "res = np.empty(thetas.shape)\n",
    "\n",
    "for i, theta in enumerate(thetas):\n",
    "    f = lambda x: ((x / (theta**2)) * np.exp(-x / theta))\n",
    "    score = 1\n",
    "    for sample in samples:\n",
    "        score *= f(sample)\n",
    "    \n",
    "    res[i] = score\n",
    "\n",
    "maxtheta = thetas[np.argmax(res)]\n",
    "print(f\"argmax: {maxtheta}\")\n",
    "plt.plot(thetas, res)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Problem 3\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [],
   "source": [
    "radial = np.array([5.2, 5.7, 7.6, 8.0, 7.7, 5.5, 6.7, 7.0, 8.4, 5.9])\n",
    "\n",
    "belted = np.array([5.1, 5.9, 7.2, 7.9, 7.8, 5.4, 6.7, 6.8, 7.9, 5.7])\n",
    "belted_mean = np.mean(belted)\n",
    "N = len(belted)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Part A\n",
    "\n",
    "Here we don't know the varience, and will therefore have to use the t distribution method.\n",
    "The interval is given by:\n",
    "$$\n",
    "\\bar X \\pm t_{\\alpha/2,n-1} \\frac S {\\sqrt{n}}\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Then the interval becomes [6.174311863003706, 7.105688136996291]\n"
     ]
    }
   ],
   "source": [
    "alpha = 0.1\n",
    "# Lets start by calculating the t value\n",
    "tval = stats.t.ppf(1 - alpha, N-1)\n",
    "\n",
    "S = np.sqrt(np.sum((belted - belted_mean)**2) / (N - 1))\n",
    "\n",
    "diff = tval * S / np.sqrt(N)\n",
    "print(f\"Then the interval becomes [{belted_mean - diff}, {belted_mean + diff}]\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Part B\n",
    "\n",
    "We can take the difference of the two and say that this should be less than or equal to $0$.\n",
    "Thus the $H_0$ is $(\\mu_{belted} - \\mu_{radial}) \\leq 0$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.1 -0.2  0.4  0.1 -0.1  0.1  0.   0.2  0.5  0.2] 0.12999999999999998\n",
      "ts: 1.9475670608117976\n",
      "p-value: 0.04164574347828587\n",
      "H_0 is accepted, thus radial does not make it better\n"
     ]
    }
   ],
   "source": [
    "mu_0 = 0\n",
    "alpha = 0.05\n",
    "diff = radial - belted\n",
    "diff_mean = np.mean(diff)\n",
    "print(diff, diff_mean)\n",
    "\n",
    "S = np.sqrt(np.sum((diff - diff_mean)**2) / (N - 1))\n",
    "\n",
    "ts = np.sqrt(N) * (diff_mean - mu_0) / S\n",
    "print(f\"ts: {ts}\")\n",
    "p_value = 1 - stats.t.cdf(ts, N - 1)\n",
    "print(f\"p-value: {p_value}\")\n",
    "\n",
    "if p_value > alpha:\n",
    "    print(\"H_0 is rejected thus radial tires have better economy\")\n",
    "else:\n",
    "    print(\"H_0 is accepted, thus radial does not make it better\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Part C\n",
    "\n",
    "Drivers can drive radically different in the way they brake, shift gears and apply power.\n",
    "The driver is therefore as important a parameter as the car driven.\n",
    "\n",
    "When creating such tests it is very important to only change the variable in question, in this case the tires.\n",
    "By keeping all other variables the same, one can say that a change in performance likely comes from the other tires."
   ]
  }
 ],
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   "display_name": "Python 3",
   "language": "python",
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