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   "source": [
    "# Learning goals\n",
    "\n",
    "Understand the following:\n",
    "\n",
    "- Probability distribution\n",
    "- Probability density function (PDF)\n",
    "- Gaussian distribution\n",
    "- Histogram\n",
    "- Mean\n",
    "- Standard deviation\n",
    "- Variance\n",
    "- Mode\n",
    "\n",
    "We will discuss these as we work through this example"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import scipy.stats as stats"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "sample_size = 30\n",
    "\n",
    "true_mean = 10.0\n",
    "true_variance = 4.0\n",
    "true_std = np.sqrt(true_variance)\n",
    "\n",
    "rng = np.random.RandomState(123)\n",
    "fish_sample = rng.normal(loc=true_mean, scale=true_std, size=sample_size)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fish_sample"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x=plt.hist(fish_sample)\n",
    "plt.xlabel(\"fish length (cm)\")\n",
    "plt.ylabel(\"counts\");"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## What is the mean of the sample of fish lengths?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "sample_mean = np.mean(fish_sample)\n",
    "print(sample_mean)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## What is the [mode](https://en.wikipedia.org/wiki/Mode_(statistics)) of the sample of fish lengths? (\"[modus](https://de.wikipedia.org/wiki/Modus_(Statistik))\" in German.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## What is the sample variance of fish lengths?\n",
    "\n",
    "We put the result in the variable `length_variance`. We use the `ddof=1` keyword argument to `np.var()` because we are estimating variance from a sample here."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "sample_variance = np.var(fish_sample, ddof=1)\n",
    "print(sample_variance)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Probability Density Functions (PDFs)\n",
    "\n",
    "- for working with continuous variables (vs. probability mass functions for discrete variables)\n",
    "- here, the area under the curve gives the probability (in contrast to probability mass functions where we have probabilities for every single value)\n",
    "- the area under the whole curve is 1. In other words, some outcome must happen.\n",
    "\n",
    "## Normal Distribution (Gaussian Distribution)\n",
    "\n",
    "- unimodal and symmetric\n",
    "- many algorithms in machine learning & statistics have normality assumptions\n",
    "- two parameters: mean (center of the peak) and standard deviation (spread); $N(\\mu, \\sigma)$\n",
    "- we can estimate parameters $\\mu$ and $\\sigma$ by the sample mean ($\\bar{x})$ and sample variance ($s^2$)\n",
    "- univariate Normal distribution:\n",
    "\n",
    "$$N(\\mu, \\sigma) = f(x \\mid \\mu, \\sigma^2) = \\frac{1}{\\sqrt{2\\pi\\sigma^2}} \\; \\exp\\bigg(-\\frac{(x-\\mu)^2}{2\\sigma^2}\\bigg)$$\n",
    "\n",
    "- standard normal distribution with zero mean and unit variance, $N(0, 1)$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def univariate_gaussian_pdf(x, mean, variance):\n",
    "    return (1. / np.sqrt(2*np.pi*variance) * \n",
    "            np.exp(- ((x - mean)**2 / (2.*variance))))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "mean = 0\n",
    "stddev = 1\n",
    "x = np.arange(-5, 5, 0.01)\n",
    "y = univariate_gaussian_pdf(x, mean, stddev**2)\n",
    "plt.plot(x, y)\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('Probability Density Function (PDF)')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "mean = 10\n",
    "stddev = 2.0\n",
    "x = np.arange(5, 15, 0.01)\n",
    "y = univariate_gaussian_pdf(x, mean, stddev**2)\n",
    "plt.plot(x, y)\n",
    "plt.xlabel('fish length (cm)')\n",
    "plt.ylabel('Probability Density Function (PDF)')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.hist(fish_sample, density=True, label='samples')\n",
    "\n",
    "x = np.linspace(5.0, 15.0, 100)\n",
    "y = univariate_gaussian_pdf(x, true_mean, true_variance)\n",
    "plt.plot(x, y, lw=5, label='theoretical (mean=%.1f, variance=%.1f)'%(true_mean,true_variance))\n",
    "\n",
    "sample_mean = np.mean(fish_sample)\n",
    "sample_variance = np.var(fish_sample, ddof=1) # for a sample variance, always use ddof=1. Here is one explanation why https://mortada.net/computing-sample-variance-why-divide-by-n-1.html\n",
    "y2 = univariate_gaussian_pdf(x, sample_mean, sample_variance)\n",
    "plt.plot(x, y2, label='empirical (mean=%.1f, variance=%.1f)'%(sample_mean,sample_variance))\n",
    "\n",
    "plt.xlabel(\"length (cm)\")\n",
    "plt.ylabel(\"probability\");\n",
    "plt.legend();\n",
    "# plt.savefig('fish-gaussian.png');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Discussion points\n",
    "\n",
    "Here are a few questions you should be able to answer:\n",
    "\n",
    " - what is plotted on the x axis?\n",
    " - what is plotted on the y axis?\n",
    " - what is the area under the curve from -∞ to ∞?\n",
    " - according to our Guassian distribution, what is the most likely fish length?\n",
    " - according to our histogram, what is the most likely fish length?"
   ]
  }
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