{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "ba322e0b-5b8f-473f-90c3-dbfb10578b34",
   "metadata": {},
   "source": [
    "## Physics 310\n",
    "\n",
    "### Class 2\n",
    "\n",
    "### Tuesday January 23, 2024\n",
    "\n",
    "\n",
    "Goals:\n",
    "* Introduce tools and functions for plotting\n",
    "* Introduce tools for generating random numbers and histograms\n",
    "* Experiment with PDFs (probability distribution functions)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "61460d81-be45-42ae-95b9-b8383d2643bb",
   "metadata": {},
   "outputs": [],
   "source": [
    "## Import relevant packages\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "from scipy import stats\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7581bf99-2e1e-4838-8002-43502dc64640",
   "metadata": {},
   "source": [
    "## The linspace function\n",
    "\n",
    "Often, we will fine that we need to generate a regularly spaced array of numbers, such as when creating an x-axis, or a set of independent variables\n",
    "\n",
    "The ```np.linspace```  is really valuable in this regard. What does it do? Look up its syntax by running the following command\n",
    "\n",
    "```?np.linspace```\n",
    "\n",
    "Run the following commands\n",
    "```np.linspace(0,30,10)``` and note what each varible does"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "432b4c1e-8077-43f2-af83-cec2d69f6805",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "227b61c8-95ce-467f-9140-9f29bf77a54b",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "5da10776-8829-45e2-9df7-b285be704bb5",
   "metadata": {},
   "source": [
    "Two possible functions that we can use for geneerating random numbers are\n",
    "\n",
    "```\n",
    "np.random.random.rand_int\n",
    "```\n",
    "and\n",
    "```\n",
    "np.random.random_sample\n",
    "```\n",
    "\n",
    "Look up the help file to determine the syntax of these two functions\n",
    "* What do the functions take as input variables\n",
    "* Do the functions have any optional variables? How do you know\n",
    "* Use the function to generate an array of 10 random numbers\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "5e02d387-04af-43f8-8e6f-0b4684308a6f",
   "metadata": {},
   "outputs": [],
   "source": [
    "#?np.random.random_sample"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f11da288-e657-4883-9951-97fdab20b7ee",
   "metadata": {},
   "source": [
    "### Gaussian Distribution\n",
    "\n",
    "This is given by \n",
    "\n",
    "$p(x) = \\frac{1}{\\sigma \\sqrt{2\\pi}}exp\\left[-\\frac{(x-\\bar{x})^2}{2\\sigma^2}\\right]$\n",
    "\n",
    "The snippet of code below generates 10 samples from a normal distribution\n",
    "```\n",
    "# Sampling from normal distribution\n",
    "n = 10\n",
    "mean = 10.\n",
    "sigma = 2.\n",
    "stats.norm.rvs(mean, sigma, size=n)  \n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "e4064506-c753-4306-b078-9317b6611e24",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Sampling from normal distribution\n",
    "n = 1000\n",
    "mean = 10.\n",
    "sigma = 2.\n",
    "ynorm = stats.norm.rvs(mean, sigma, size=n)  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "6083e6e1-b675-4f95-979a-4af96d13d290",
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.linspace(0,20,200)  # make an array of 200 evenly spaced values from 0 to 20\n",
    "y = stats.norm.pdf(x, mean, sigma)      # determine the value of the pdf at each of the points in 'x'\n",
    "plt.title(\"pdf of normal distribution\")\n",
    "plt.xlabel(\"$x$\")\n",
    "plt.ylabel(\"$p(x)$\")\n",
    "plt.grid()\n",
    "plt.plot(x, y);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4d756685-2053-4558-a0ee-6804b7ed12a9",
   "metadata": {},
   "source": [
    "Consider the following snippet of code\n",
    "```\n",
    "n = 1000\n",
    "mean = 10.\n",
    "sigma = 2.\n",
    "ynorm = stats.norm.rvs(mean, sigma, size=n)  \n",
    "```\n",
    "\n",
    "Before runnint it, describe what it will do.\n",
    "\n",
    "Now generate a histogram of using:\n",
    "\n",
    "```\n",
    "plt.hist(ynorm)\n",
    "plt.xlabel(\"value\")\n",
    "plt.ylabel(\"occurences\")\n",
    "plt.title(\"Histogram; equal sized bins\")\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c98073ac-0ad0-49ba-9902-8f8022465a6f",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "b3f0171a-20de-4e8c-89ed-0c00b87971bb",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "dffb4027-bc7b-427e-ab5c-ed20469421ca",
   "metadata": {},
   "source": [
    "Now the plt.hist function has a number of optional variables. For example, you can normalize it\n",
    "using density = True, and you can change the relative width of the cells by using rwidth = ...\n",
    "\n",
    "Try this snippet of code, and think about what each line an each optional variable does\n",
    "\n",
    "```\n",
    "plt.hist(ynorm, density = True,rwidth=0.8,label = 'samples')\n",
    "plt.plot(x, y, label = 'pdf');\n",
    "plt.legend()\n",
    "```\n",
    "\n",
    "How would you add labels on the x and y axes?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "661cefba-0737-4e22-a0e6-cf821808c7e7",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "9797febf-2e56-4d99-bc9a-f2ac2bc4a54a",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "9c959bd3-89c9-4d24-8a9a-7496ac645c7e",
   "metadata": {},
   "source": [
    "## Using other distributions\n",
    "\n",
    "The stats module implements dozens of other distributions. For example, to generate random values from a binomial distribution, you may use\n",
    "\n",
    "```stats.binom.rvs(n,p,size = 100)``` where ```n``` and ```p``` are the two required variables for the binomial distribution. \n",
    "\n",
    "```stats.poisson.rvs(mean, size=100)``` generates a sample of 100 random variables form the poisson distribution. Note that the poisson distribution is a function of a single variable. Namely the mean number of counts.\n",
    "\n",
    "Experiment with generating random numbers and histograms of these distributions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "18013a25-2dd6-456a-b29a-31259ddca19d",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "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": "3.10.9"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}