manual_teachbooks/book/examples/example_gumbel_stripped.ipynb

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    "## Gumbel Distribution Exercise\n",
    "\n",
    "Imagine you are concerned with the concentration of an airborne contaminant, $X$, measured in ppm. A previous benchmark study estimates that there is a 10% and 1% chance, respectively, of exceeding 4 and 10 ppm, respectively. You have been asked by the regulatory agency to estimate the contaminant concentration with 0.1% probability of being exceeded. Prior studies suggest that a Gumbel distribution can be used to model contaminant concentration, given by the CDF:\n",
    "\n",
    "$$\n",
    "F_X(x) = \\exp{\\Bigg( -\\exp{\\Big(-\\frac{x-\\mu}{\\beta}\\Big)}\\Bigg)}\n",
    "$$\n",
    "\n",
    "Using the cell blocks below as a guide (and also to check your analysis): Task 1) find the parameters of a Gumbel distribution that matches the information provided above, then, Tasks 2-3) use it to estimate the concentration with exceedance probability 0.1%.\n",
    "\n",
    "To complete this assignment, you can use `numpy`, `matplotlib` and from the `math` library, `log` and `e`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "from math import log, e\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Task 0:** fill in the appropriate fitting points:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "x_1 = _\n",
    "p_1 = _\n",
    "x_2 = _\n",
    "p_2 = _"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Task 1:** derive the distribution parameters:"
   ]
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   "source": [
    "def gumbel_2_points(x_1, p_1, x_2, p_2): \n",
    "    \"\"\"Compute Gumbel distribution parameters from two points of the CDF.\n",
    "        Arguments:\n",
    "            x_1 (float): point one\n",
    "            p_1 (float): cumulative probability for point 1\n",
    "            x_2 (float): point two\n",
    "            p_2 (float): cumulative probability for point 2\n",
    "    \"\"\"\n",
    "\n",
    "    # YOUR CODE GOES HERE #\n",
    "    beta = _\n",
    "    mu = _\n",
    "    #######################\n",
    "\n",
    "    return mu, beta"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The cells below will print your parameter values and create a plot to help confirm you have the right implementation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "tags": []
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   "outputs": [],
   "source": [
    "def plot_distribution(mu, beta,\n",
    "                      x_1, p_1, x_2, p_2):\n",
    "    plt.title(\"Gumbel Distribution, $1-F_X(x)$\")\n",
    "    plt.xlabel(\"Contaminant Concentration, $X$ [ppm]\")\n",
    "    plt.ylabel(\"Exceedance Probability [--]\")\n",
    "    plt.grid(color='black', linestyle='-', linewidth=0.3)\n",
    "    x_axis = np.arange(0, 20, 0.1)\n",
    "    plt.plot(x_axis, np.vectorize(gumbel_distribution)(x_axis), linewidth=2)\n",
    "    plt.plot(x_1, 1 - p_1, 'ro')\n",
    "    plt.annotate(\"Point 1\", (x_1 + 0.4, 1 - p_1 + 0.02))\n",
    "    plt.plot(x_2, 1 - p_2, 'ro')\n",
    "    plt.annotate(\"Point 2\", (x_2 + 0.4, 1 - p_2 + 0.001))\n",
    "    plt.yscale(\"log\")   \n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "mu, beta = gumbel_2_points(x_1, p_1, x_2, p_2)\n",
    "\n",
    "print(f\"Your mu: {mu:0.5f}\\nYour beta: {beta:0.5f}\")\n",
    "\n",
    "gumbel_distribution = lambda x: 1 - e**(-e**(-(x - mu)/beta))\n",
    "\n",
    "plot_distribution(mu, beta, x_1, p_1, x_2, p_2)\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Task 2:** write a function to solve for the random variable value (it will be tested for you with the Check answer button, along with the distribution parameters)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def find_x_with_probability_p(p): \n",
    "    \"\"\" Compute point in the gumbel distribution for which the CDF is p\n",
    "        Use the variables mu and beta defined above!\n",
    "        Hint: they have been defined globally, so you don't need to \n",
    "                include them as arguments.\n",
    "    \"\"\"\n",
    "    # YOUR CODE GOES HERE #\n",
    "    x = _\n",
    "    #######################\n",
    "\n",
    "    return x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Task 3:** use the function `find_x_with_probability_p` to evaluate the random variable value with exceedance probabiltiy of 0.001."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# YOUR CODE GOES HERE #\n",
    "x = _\n",
    "#######################\n",
    "\n",
    "print(f\"The value of x with probability of exceedance 0.001 os {x:0.5f}\")"
   ]
  }
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