joint_dist_X_abs_X_normal_dist.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### We want to calculate Joint Distribution of $X$ and $|X|$ When $X \\sim \\mathcal{N}(0, \\sigma^2)$\n",
    "<hr>\n",
    "<br><br>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{aligned}\n",
    "& X\\sim\\mathcal{N}(0, \\sigma^2), \\quad Y = \\vert X \\vert \\implies \\\\\n",
    "& P^* = \\mathcal{P}(X \\leq x, Y \\leq y) = F_{X, Y}(x, y) = ? \\\\\n",
    "& \\mathcal{P}(X \\leq x, Y \\leq y) = \\mathcal{P}(X \\leq x, Y \\leq y, X > 0) + \\mathcal{P}(X \\leq x, Y \\leq y, X < 0)\\\\\n",
    "& \\mathcal{P}(0 < X \\leq x, ~ X \\leq y) + \\mathcal{P}(X \\leq \\min(0, x), ~0 < -X < y) \\implies \\\\\n",
    "& P^* = \\mathcal{P}(0 < X \\leq \\min(x, y)) + \\mathcal{P}(X \\leq \\min(0, x), -y < X < 0) \\implies \\\\\n",
    "& P^* = \\begin{cases}\n",
    "x > 0: & \\mathcal{P}(-y \\leq X \\leq \\min(x, y)) \\\\\n",
    "x < 0: & \\begin{cases} \n",
    "x > -y: & \\mathcal{P}(-y < X < x) \\\\\n",
    "x < -y: & 0.\n",
    "\\end{cases}\n",
    "\\end{cases}\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br> \n",
    "\n",
    "### To show that the way we followed was the correct method. We also estimate the results based on the simulation and compare it to the above proposition.\n",
    "\n",
    "\n",
    "```python code ```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "# load require libraries \n",
    "from scipy import stats\n",
    "import numpy as np\n",
    "import pandas as pd \n",
    "\n",
    "class get_prob():\n",
    "    def __init__(self, nsim):\n",
    "        ## define number of simulation\n",
    "        self.nsim = nsim\n",
    "        \n",
    "        # define method for get x and y\n",
    "    def get_xy(self, x = 1, y = 2, sigg = 2, seed = 123): \n",
    "        self.x = x\n",
    "        self.y = y \n",
    "        self.sigg = sigg\n",
    "        self.seed = seed\n",
    "        \n",
    "    ## define method for implement simulation \n",
    "    def sim_main(self): \n",
    "        sim_res = stats.norm.rvs(loc = 0, scale = self.sigg, size = self.nsim, random_state = self.seed)\n",
    "        temp1 = list(map(lambda u: 1 if (u <= self.x and np.abs(u) <= self.y) else 0, sim_res))\n",
    "        return np.mean(temp1)\n",
    "    \n",
    "    ## define method for exactly value \n",
    "    def exact_prob(self): \n",
    "        rvss = stats.norm(loc = 0, scale = self.sigg) \n",
    "        if self.x > 0: \n",
    "            temp = (rvss.cdf(np.min([self.x, self.y])) - rvss.cdf(-self.y))\n",
    "        elif self.x <= -self.y: \n",
    "            temp = 0\n",
    "        else: \n",
    "            temp = (rvss.cdf(self.x) - rvss.cdf(-self.y))\n",
    "        return temp \n",
    "    \n",
    "    # return results \n",
    "    def output_result(self): \n",
    "        temp2 = {\"nsim\": self.nsim, \"x\": self.x, \"y\": self.y, \"Sig\": self.sigg,\n",
    "                 \"sim_result\": self.sim_main(), \"Exact_result\": self.exact_prob()}\n",
    "        temp3 = pd.DataFrame(temp2, index = [1])\n",
    "        return temp3 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [],
   "source": [
    "nList = list(map(lambda x: int(x), [1e+4, 1e+5]))\n",
    "xList = [-2, -1, 1, 2]\n",
    "yList = [.5, 1.5, 2.5, 3.5]\n",
    "sigList = [0.1, 1.5, 4]\n",
    "Result = []\n",
    "for i in sigList: \n",
    "    for j in xList: \n",
    "        for k in yList: \n",
    "            for m in nList:\n",
    "                Prob = get_prob(nsim = m)\n",
    "                Prob.get_xy(x = j, y = k, sigg = i)\n",
    "                tempp = Prob.output_result()\n",
    "                Result.append(tempp) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000 -2  0.5  0.1         0.0             0 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000 -2  0.5  0.1         0.0             0 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000 -2  1.5  0.1         0.0             0 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000 -2  1.5  0.1         0.0             0 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000 -2  2.5  0.1         0.0  2.753624e-89 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000 -2  2.5  0.1         0.0  2.753624e-89 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000 -2  3.5  0.1         0.0  2.753624e-89 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000 -2  3.5  0.1         0.0  2.753624e-89 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000 -1  0.5  0.1         0.0             0 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000 -1  0.5  0.1         0.0             0 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000 -1  1.5  0.1         0.0  7.619853e-24 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000 -1  1.5  0.1         0.0  7.619853e-24 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000 -1  2.5  0.1         0.0  7.619853e-24 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000 -1  2.5  0.1         0.0  7.619853e-24 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000 -1  3.5  0.1         0.0  7.619853e-24 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000 -1  3.5  0.1         0.0  7.619853e-24 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000  1  0.5  0.1         1.0      0.999999 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000  1  0.5  0.1         1.0      0.999999 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000  1  1.5  0.1         1.0           1.0 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000  1  1.5  0.1         1.0           1.0 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000  1  2.5  0.1         1.0           1.0 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000  1  2.5  0.1         1.0           1.0 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000  1  3.5  0.1         1.0           1.0 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000  1  3.5  0.1         1.0           1.0 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000  2  0.5  0.1         1.0      0.999999 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000  2  0.5  0.1         1.0      0.999999 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000  2  1.5  0.1         1.0           1.0 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000  2  1.5  0.1         1.0           1.0 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000  2  2.5  0.1         1.0           1.0 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000  2  2.5  0.1         1.0           1.0 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000  2  3.5  0.1         1.0           1.0 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000  2  3.5  0.1         1.0           1.0 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000 -2  0.5  1.5         0.0             0 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000 -2  0.5  1.5         0.0             0 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000 -2  1.5  1.5         0.0             0 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000 -2  1.5  1.5         0.0             0 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000 -2  2.5  1.5      0.0485      0.043421 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000 -2  2.5  1.5     0.04401      0.043421 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000 -2  3.5  1.5      0.0836      0.081396 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000 -2  3.5  1.5     0.08112      0.081396 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000 -1  0.5  1.5         0.0             0 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000 -1  0.5  1.5         0.0             0 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000 -1  1.5  1.5      0.0951      0.093837 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000 -1  1.5  1.5     0.09471      0.093837 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000 -1  2.5  1.5      0.2078      0.204702 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000 -1  2.5  1.5     0.20657      0.204702 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000 -1  3.5  1.5      0.2429      0.242677 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000 -1  3.5  1.5     0.24368      0.242677 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000  1  0.5  1.5      0.2626      0.261117 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000  1  0.5  1.5     0.26085      0.261117 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000  1  1.5  1.5      0.5877      0.588852 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000  1  1.5  1.5     0.58863      0.588852 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000  1  2.5  1.5      0.7004      0.699717 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000  1  2.5  1.5     0.70049      0.699717 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000  1  3.5  1.5      0.7355      0.737692 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000  1  3.5  1.5      0.7376      0.737692 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000  2  0.5  1.5      0.2626      0.261117 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000  2  0.5  1.5     0.26085      0.261117 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000  2  1.5  1.5      0.6843      0.682689 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000  2  1.5  1.5     0.68124      0.682689 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000  2  2.5  1.5      0.8641      0.860998 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000  2  2.5  1.5     0.86187      0.860998 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000  2  3.5  1.5      0.8992      0.898973 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000  2  3.5  1.5     0.89898      0.898973 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000 -2  0.5    4         0.0             0 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000 -2  0.5    4         0.0             0 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000 -2  1.5    4         0.0             0 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000 -2  1.5    4         0.0             0 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000 -2  2.5    4      0.0426      0.042552 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000 -2  2.5    4     0.04187      0.042552 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000 -2  3.5    4      0.1181      0.117751 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000 -2  3.5    4     0.11763      0.117751 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000 -1  0.5    4         0.0             0 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000 -1  0.5    4         0.0             0 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000 -1  1.5    4      0.0437      0.047463 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000 -1  1.5    4     0.04703      0.047463 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000 -1  2.5    4      0.1294      0.135308 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000 -1  2.5    4     0.13405      0.135308 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000 -1  3.5    4      0.2049      0.210507 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000 -1  3.5    4     0.20981      0.210507 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000  1  0.5    4      0.0973      0.099476 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000  1  0.5    4     0.09946      0.099476 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000  1  1.5    4      0.2418      0.244876 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000  1  1.5    4     0.24445      0.244876 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000  1  2.5    4      0.3275      0.332721 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000  1  2.5    4     0.33147      0.332721 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000  1  3.5    4       0.403      0.407919 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000  1  3.5    4     0.40723      0.407919 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000  2  0.5    4      0.0973      0.099476 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000  2  0.5    4     0.09946      0.099476 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000  2  1.5    4      0.2913       0.29234 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000  2  1.5    4     0.29223       0.29234 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000  2  2.5    4      0.4193      0.425477 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000  2  2.5    4     0.42348      0.425477 \n",
      "\n",
      "\n",
      "\n",
      "    nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  10000  2  3.5    4      0.4948      0.500676 \n",
      "\n",
      "\n",
      "\n",
      "     nsim  x    y  Sig  sim_result  Exact_result\n",
      "1  100000  2  3.5    4     0.49924      0.500676 \n",
      "\n",
      "\n",
      "\n"
     ]
    }
   ],
   "source": [
    "for k in Result: \n",
    "    print(k, \"\\n\\n\\n\")"
   ]
  }
 ],
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