Trigonometric_transformation_random_variable_uniform_distribution.ipynb

{
 "cells": [
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### $\\text{X} \\sim \\text{Unif}(0, ~1) \\to\\quad \\text{if}~~ ~ Y = \\sin(2\\pi X)\\to \\text{denstiy of }~~Y = ?$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from sympy import*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy.stats import Uniform, density, cdf, E, variance, ContinuousRV"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = Uniform(\"x\", 0, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "pi = Symbol(\"pi\", positive=True)\n",
    "x, y = symbols('x y')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\begin{cases} 1 & \\text{for}\\: x \\geq 0 \\wedge x \\leq 1 \\\\0 & \\text{otherwise} \\end{cases}$"
      ],
      "text/plain": [
       "Piecewise((1, (x >= 0) & (x <= 1)), (0, True))"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "density(X)(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\begin{cases} 0 & \\text{for}\\: x < 0 \\\\x & \\text{for}\\: x \\leq 1 \\\\1 & \\text{otherwise} \\end{cases}$"
      ],
      "text/plain": [
       "Piecewise((0, x < 0), (x, x <= 1), (1, True))"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cdf(X)(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{1}{2}$"
      ],
      "text/plain": [
       "1/2"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{1}{12}$"
      ],
      "text/plain": [
       "1/12"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "variance(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\begin{cases} 0 & \\text{for}\\: \\sin{\\left(2 \\pi y \\right)} < 0 \\\\\\sin{\\left(2 \\pi y \\right)} & \\text{for}\\: \\sin{\\left(2 \\pi y \\right)} \\leq 1 \\\\1 & \\text{otherwise} \\end{cases}$"
      ],
      "text/plain": [
       "Piecewise((0, sin(2*pi*y) < 0), (sin(2*pi*y), sin(2*pi*y) <= 1), (1, True))"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cdf(X)(sin(2 * pi * y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{1}{\\pi \\sqrt{1 - y^{2}}}$"
      ],
      "text/plain": [
       "1/(pi*sqrt(1 - y**2))"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pDf = 1/pi * 1/sqrt(1 - y**2)\n",
    "pDf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\operatorname{asin}{\\left(y \\right)}}{\\pi}$"
      ],
      "text/plain": [
       "asin(y)/pi"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(pDf, y)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br><br><br><br>\n",
    "<br><b>\n",
    "<div style = \"color:red;\">\n",
    "\n",
    "#### Create A Trigonometric Circle for explain Problem \n",
    "#### define a class for this issue...\n",
    "<br><br></b></div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "from mpl_toolkits.axisartist import SubplotZero\n",
    "\n",
    "\n",
    "class TrigAxes():\n",
    "    \n",
    "    def __init__(self, figsize=(12,5)):\n",
    "        self.xlim = (-1.2, 1.2)\n",
    "        self.ylim = (-1.2, 1.2)\n",
    "        self.figsize  = figsize\n",
    "        self.angles   = []\n",
    "        self.scale_arrows()\n",
    "\n",
    "    def __arrow__(self, x, y, dx, dy, width, length):\n",
    "        plt.arrow(\n",
    "            x, y, dx, dy, \n",
    "            color       = 'k',\n",
    "            clip_on     = False, \n",
    "            head_width  = self.head_width, \n",
    "            head_length = self.head_length\n",
    "        ) \n",
    "        \n",
    "    def __drawAxis__(self):\n",
    "        \"\"\"\n",
    "        Draws the 2D cartesian axis\n",
    "        \"\"\"\n",
    "        # A subplot with two additional axis, \"xzero\" and \"yzero\"\n",
    "        # corresponding to the cartesian axis\n",
    "        ax = SubplotZero(self.fig, 1, 1, 1)\n",
    "        self.fig.add_subplot(ax)\n",
    "        \n",
    "        # make xzero axis (horizontal axis line through y=0) visible.\n",
    "        for axis in [\"xzero\",\"yzero\"]:\n",
    "            ax.axis[axis].set_visible(True)\n",
    "        # make the other axis (left, bottom, top, right) invisible\n",
    "        for n in [\"left\", \"right\", \"bottom\", \"top\"]:\n",
    "            ax.axis[n].set_visible(False)\n",
    "            \n",
    "        # Plot limits\n",
    "        plt.xlim(self.xlim)\n",
    "        plt.ylim(self.ylim)\n",
    "        \n",
    "        # No ticks for the trigonometric circle\n",
    "        ax = plt.gca()\n",
    "        ax.axes.xaxis.set_ticks([])\n",
    "        ax.axes.yaxis.set_ticks([])\n",
    "\n",
    "        # Draw the trigonometric circle\n",
    "        self.circle = plt.Circle((0, 0), 1, color='#000000', fill=False)\n",
    "        ax.add_artist(self.circle)\n",
    "        \n",
    "        # Draw the arrows\n",
    "        self.__arrow__(self.xlim[1], 0, 0.01, 0, 0.3, 0.2) # x-axis arrow\n",
    "        self.__arrow__(0, self.ylim[1], 0, 0.01, 0.2, 0.3) # y-axis arrow\n",
    "        \n",
    "    def scale_arrows(self):\n",
    "        \"\"\" Make the arrows look good regardless of the axis limits \"\"\"\n",
    "        xrange = self.xlim[1] - self.xlim[0]\n",
    "        yrange = self.ylim[1] - self.ylim[0]\n",
    "        \n",
    "        self.head_width  = min(xrange/30, 0.25)\n",
    "        self.head_length = min(yrange/30, 0.3)\n",
    "        \n",
    "        \n",
    "    def draw(self, imgpath=None):\n",
    "        \n",
    "        self.scale_arrows()\n",
    "        self.fig = plt.figure(figsize=self.figsize)\n",
    "        # First draw the axis\n",
    "        self.__drawAxis__()\n",
    "        # Plot each vector\n",
    "        for angle in self.angles:\n",
    "            angle.draw()\n",
    "        # Save the image?\n",
    "        if imgpath:\n",
    "            plt.tight_layout()\n",
    "            plt.savefig(imgpath, dpi=100)\n",
    "        plt.show()\n",
    "    \n",
    "    #-------------------------------------------------\n",
    "    # Angles\n",
    "    #-------------------------------------------------\n",
    "    def addAngles(self, angles):\n",
    "        for a in angles:\n",
    "            self.addAngle(a)\n",
    "            \n",
    "    def addAngle(self, a):\n",
    "        # Modify the head length & width of the vector \n",
    "        # so it looks good for this axis\n",
    "        a.head_width  =  self.head_width / 2.\n",
    "        a.head_length =  self.head_length / 2.\n",
    "        self.angles.append(a)\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "## define a class for add angle to circle of Trigonometric\n",
    "\n",
    "class Angle():\n",
    "    \"\"\"\n",
    "    Angle(theta, label)\n",
    "        theta: angle on the trigonometric circle\n",
    "        label: optional label, default is theta\n",
    "    \"\"\"\n",
    "    def __init__(self, theta, \n",
    "                 label       = None,\n",
    "                 color       = '#6897bb', \n",
    "                 linewidth   = 0.01, \n",
    "                 linestyle   = '-',\n",
    "                 head_width  = 0.25,\n",
    "                 head_length = 0.3\n",
    "                ):\n",
    "        # Theta, label\n",
    "        self.theta = math.radians(theta)\n",
    "        if label:\n",
    "            self.label = str(label)\n",
    "        else:\n",
    "            self.label = str(theta)\n",
    "            \n",
    "        # Drawing options\n",
    "        self.color = color\n",
    "        self.linewidth = linewidth\n",
    "        self.linestyle = linestyle\n",
    "        self.head_width  = head_width\n",
    "        self.head_length = head_length\n",
    "        self.text_args = {\n",
    "            'ha': 'left', \n",
    "            'va': 'center', \n",
    "            'fontsize': 12, \n",
    "            'color': self.color\n",
    "        }\n",
    "\n",
    "    \n",
    "    def draw(self):\n",
    "        # Plot the vector\n",
    "        plt.arrow(\n",
    "            0, 0,\n",
    "            math.cos(self.theta), math.sin(self.theta), \n",
    "            color = self.color,\n",
    "            width = self.linewidth,\n",
    "            head_width = self.head_width, \n",
    "            head_length = self.head_length\n",
    "        )\n",
    "        # Add the label\n",
    "        plt.text(\n",
    "            math.cos(self.theta) + 0.1,\n",
    "            math.sin(self.theta) + 0.1,\n",
    "            self.label,\n",
    "            **self.text_args\n",
    "        )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 500x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Create the trigonometric axis\n",
    "axes = TrigAxes(figsize=(5,4))\n",
    "\n",
    "axes.addAngle(Angle(30, label='theta1 = 30', color='blue', linewidth=0.02))\n",
    "axes.addAngle(Angle(150, label='theta2 = 150', color='red', linewidth=0.02))\n",
    "axes.addAngle(Angle(-30, label='theta3 = -30', color='green', linewidth=0.02))\n",
    "axes.addAngle(Angle(-150, label='theta4 = -150', color='brown', linewidth=0.02))\n",
    "\n",
    "axes.draw()"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br><br><br><br>\n",
    "<hr>\n",
    "<br><br>\n",
    "<div style = \"color : blue;\">\n",
    "\n",
    "#### Manual Soloution"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{align}\n",
    "& \\mathcal{P}(Y \\leq y) = \\begin{cases}p_1 & 0 \\leq y \\leq 1, \\\\ p_2 & -1 \\leq y < 0. \\end{cases} \\implies \\\\\n",
    "& p_1 = \\mathcal{P}(Y \\leq 0) + \\mathcal{P}(Y \\leq y, y > 0) = \\mathcal{P}\\left(\\sin(2\\pi X) \\leq y, x \\leq \\frac{1}{2}\\right) = \\\\\n",
    "& \\mathcal{P}(Y \\leq 0) + \\mathcal{P}\\left(0 \\leq x \\leq \\frac{\\arcsin(y)}{2\\pi}\\right) + \\mathcal{P}\\left(\\frac{\\pi - \\arcsin(y)}{2\\pi} \\leq x \\leq \\frac{\\pi}{2\\pi}\\right)= \\\\\n",
    "& \\mathcal{P}(Y \\leq 0) + \\frac{\\arcsin(y)}{2\\pi} + \\frac{\\pi}{2\\pi} - \\frac{\\pi - \\arcsin(y)}{2\\pi} = \\mathcal{P}(Y \\leq 0) + \\frac{\\arcsin(y)}{\\pi}, \\\\\n",
    "& p_2 = \\mathcal{P}(Y \\leq y, y < 0) = \\mathcal{P}\\left(\\sin(2\\pi X) \\leq y, x > \\frac{1}{2}\\right) = \\\\\n",
    "& \\mathcal{P}\\left(\\frac{\\frac{3\\pi}{2}}{2\\pi} \\leq x \\leq \\frac{2\\pi + \\arcsin(y)}{2\\pi}\\right) + \\mathcal{P}\\left(\\frac{\\pi - \\arcsin(y)}{2\\pi} \\leq x \\leq \\frac{\\frac{3\\pi}{2}}{2\\pi}\\right)= \\\\\n",
    "& \\frac{2\\pi + \\arcsin(y)}{2\\pi} - \\frac{\\frac{3\\pi}{2}}{2\\pi} + \\frac{\\frac{3\\pi}{2}}{2\\pi} -  \\frac{\\pi - \\arcsin(y)}{2\\pi} = \\\\\n",
    "& \\frac{\\pi}{2\\pi} + \\frac{2\\arcsin(y)}{2\\pi} = \\frac{1}{2} + \\frac{\\arcsin(y)}{\\pi} \\implies \\\\\n",
    "& \\mathcal{P}(Y \\leq 0) = \\frac{1}{2} + \\frac{\\arcsin(0)}{\\pi} = \\frac{1}{2} \\implies \\\\\n",
    "& p_1 = \\frac{1}{2} + \\frac{\\arcsin(y)}{\\pi} \\implies \\\\\n",
    "& F_Y(y) = \\begin{cases} 0 & y < -1, \\\\ \\frac{1}{2} + \\frac{\\arcsin(y)}{\\pi} & -1 \\leq y < 1, \\\\ 1 & \n",
    "y \\geq 1.\\end{cases} \\implies \\\\\n",
    "& f_Y(y) = \\begin{cases} \\frac{1}{\\pi\\sqrt{1 - y^2}} & -1 < y < 1, \\\\ 0 & \\text{Others}. \\end{cases}\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br><br><br><hr><br><br>\n",
    "\n",
    "<div style = \"color : blue;\">\n",
    "\n",
    "#### Python Soloution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "Y = ContinuousRV(y, density = pDf, set = Interval.open(-1, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\begin{cases} \\frac{1}{\\pi \\sqrt{1 - y^{2}}} & \\text{for}\\: y > -1 \\wedge y < 1 \\\\0 & \\text{otherwise} \\end{cases}$"
      ],
      "text/plain": [
       "Piecewise((1/(pi*sqrt(1 - y**2)), (y > -1) & (y < 1)), (0, True))"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "density(Y)(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\begin{cases} \\begin{cases} 0 & \\text{for}\\: y < -1 \\\\\\frac{\\operatorname{asin}{\\left(\\min\\left(1, y\\right) \\right)}}{\\pi} + \\frac{\\pi}{2 \\pi} & \\text{otherwise} \\end{cases} & \\text{for}\\: y \\geq -1 \\\\0 & \\text{otherwise} \\end{cases}$"
      ],
      "text/plain": [
       "Piecewise((Piecewise((0, y < -1), (asin(Min(1, y))/pi + pi/(2*pi), True)), y >= -1), (0, True))"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cdf(Y)(y)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Expectation of Y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E(Y)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Variance of Y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\pi}{2 \\pi}$"
      ],
      "text/plain": [
       "pi/(2*pi)"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "variance(Y)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "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": "3.11.0"
  },
  "orig_nbformat": 4,
  "vscode": {
   "interpreter": {
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 },
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 "nbformat_minor": 2
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