Untitled9.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# non-linear equations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy\n",
    "sympy.init_printing()\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "x, a, b, c = sympy.symbols(\"x, a, b, c\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\frac{- b + \\sqrt{- 4 a c + b^{2}}}{2 c}, \\  - \\frac{b + \\sqrt{- 4 a c + b^{2}}}{2 c}\\right]$"
      ],
      "text/plain": [
       "⎡        _____________   ⎛       _____________⎞ ⎤\n",
       "⎢       ╱           2    ⎜      ╱           2 ⎟ ⎥\n",
       "⎢-b + ╲╱  -4⋅a⋅c + b    -⎝b + ╲╱  -4⋅a⋅c + b  ⎠ ⎥\n",
       "⎢─────────────────────, ────────────────────────⎥\n",
       "⎣         2⋅c                     2⋅c           ⎦"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve(a + b*x + c*x**2, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ - 2 \\operatorname{atan}{\\left(\\frac{b - \\sqrt{a^{2} + b^{2}}}{a} \\right)}, \\  - 2 \\operatorname{atan}{\\left(\\frac{b + \\sqrt{a^{2} + b^{2}}}{a} \\right)}\\right]$"
      ],
      "text/plain": [
       "⎡       ⎛       _________⎞         ⎛       _________⎞⎤\n",
       "⎢       ⎜      ╱  2    2 ⎟         ⎜      ╱  2    2 ⎟⎥\n",
       "⎢       ⎜b - ╲╱  a  + b  ⎟         ⎜b + ╲╱  a  + b  ⎟⎥\n",
       "⎢-2⋅atan⎜────────────────⎟, -2⋅atan⎜────────────────⎟⎥\n",
       "⎣       ⎝       a        ⎠         ⎝       a        ⎠⎦"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve(a * sympy.cos(x) - b * sympy.sin(x), x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "ename": "NotImplementedError",
     "evalue": "multiple generators [x, sin(x)]\nNo algorithms are implemented to solve equation -x + sin(x)",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mNotImplementedError\u001b[0m                       Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-8-d3ec18de60c6>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[0msympy\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0msolve\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0msympy\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0msin\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m-\u001b[0m\u001b[0mx\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mx\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[1;32mC:\\ProgramData\\Anaconda3\\lib\\site-packages\\sympy\\solvers\\solvers.py\u001b[0m in \u001b[0;36msolve\u001b[1;34m(f, *symbols, **flags)\u001b[0m\n\u001b[0;32m   1169\u001b[0m     \u001b[1;31m###########################################################################\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   1170\u001b[0m     \u001b[1;32mif\u001b[0m \u001b[0mbare_f\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 1171\u001b[1;33m         \u001b[0msolution\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0m_solve\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;33m*\u001b[0m\u001b[0msymbols\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;33m**\u001b[0m\u001b[0mflags\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   1172\u001b[0m     \u001b[1;32melse\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   1173\u001b[0m         \u001b[0msolution\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0m_solve_system\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0msymbols\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;33m**\u001b[0m\u001b[0mflags\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32mC:\\ProgramData\\Anaconda3\\lib\\site-packages\\sympy\\solvers\\solvers.py\u001b[0m in \u001b[0;36m_solve\u001b[1;34m(f, *symbols, **flags)\u001b[0m\n\u001b[0;32m   1740\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   1741\u001b[0m     \u001b[1;32mif\u001b[0m \u001b[0mresult\u001b[0m \u001b[1;32mis\u001b[0m \u001b[1;32mFalse\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 1742\u001b[1;33m         \u001b[1;32mraise\u001b[0m \u001b[0mNotImplementedError\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m'\\n'\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mjoin\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m[\u001b[0m\u001b[0mmsg\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mnot_impl_msg\u001b[0m \u001b[1;33m%\u001b[0m \u001b[0mf\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   1743\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   1744\u001b[0m     \u001b[1;32mif\u001b[0m \u001b[0mflags\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mget\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m'simplify'\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;32mTrue\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mNotImplementedError\u001b[0m: multiple generators [x, sin(x)]\nNo algorithms are implemented to solve equation -x + sin(x)"
     ]
    }
   ],
   "source": [
    "sympy.solve(sympy.sin(x)-x, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "from math import factorial"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 120$"
      ],
      "text/plain": [
       "120"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factorial(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "n=9\n",
    "p=0.45"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA4AAAASCAYAAABrXO8xAAAABHNCSVQICAgIfAhkiAAAAP5JREFUOI3N0r8rxVEYx/FXftUtSordYEImQplkYeW/MSol+SfYMFjkx6K7KIMySl2KkqRrkd+u4fvcup3OvZPBs3xO53Pe5/k8p8Mf1wx2cY/30CPM1w+0Z6BlbKIf+wHcYTD841ynJdTC7Mn4nTmoDRW8RLeW1dGwno44O6hiASN4wxlOm4HjoQ84x2jSpIxFPKbdVxXzfeEKs+jGMA7CO8nFXgvzG2OJV8Jt+FMUD1KvamgFFwn4isNYT6TgZehzLk7DxaUULCvmG0JXBhwJvcnduqWYYyXZn8NPpOnNgQOKF61FgnVsR5JPxc9qWn3YwDU+8IQ9TLaC/nH9AnM2OpXemhB0AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle 6$"
      ],
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s = np.random.binomial(n, p)\n",
    "s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "def prob_densB(k):\n",
    "    n=9\n",
    "    p=0.45\n",
    "    P=factorial(n) / (factorial(k) * factorial(n-k)) * p ** k * (1-p) ** (n-k)\n",
    "    return P"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 0.004605366583984379$"
      ],
      "text/plain": [
       "0.004605366583984379"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "prob_densB(0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 0.03391224484570315$"
      ],
      "text/plain": [
       "0.03391224484570315"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "prob_densB(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 0.11098552858593758$"
      ],
      "text/plain": [
       "0.11098552858593758"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "prob_densB(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 0.2118814636640626$"
      ],
      "text/plain": [
       "0.2118814636640626"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "prob_densB(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 0.26003634176953133$"
      ],
      "text/plain": [
       "0.26003634176953133"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "prob_densB(4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 0.2127570069023438$"
      ],
      "text/plain": [
       "0.2127570069023438"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "prob_densB(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 0.11604927649218756$"
      ],
      "text/plain": [
       "0.11604927649218756"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "prob_densB(6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "prob_densB(k)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "x=range(10)\n",
    "y=[prob_densB(k) for k in range(10)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x,y)\n",
    "plt.grid()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAN8AAAASCAYAAADbjwtGAAAABHNCSVQICAgIfAhkiAAAB7VJREFUaIHt2nnsXUUVB/BPa6u0UlRUaFSCSGsobdKCWhGXvlZAEUuK2x9EwLjEhQg1IArR9Bc3FhFpiksqbgQ10WAlFATUVAsVRFwpRRbbpxBoKpVCKbXY9ucfZ67v9v7ufe/d+34m/vG+ycvNmzkzc+acM2fmnBmGGGKI/xu8BN/Cw9iFNi7H82r2MwHvxe3YjqfwB5yFZ3Rp93pcg0fS+I/gZrylhPYdWIFb8ARGcXVNPk9L7Ubx/gqakxIPD2EnNuJHeE0J7fNTP6vwQKJ/HLfifZhYMcbF+AUeTG3+KeS1LPVZhTq8NaFv68in+NtcQv+eLvTZb0/FWHVsr4m86tpLE1229SmvCYWGh+PXOAjX4i+Yj4W4F6/F1i7M5nGVMOwtuA47cByOFIvrnYmhPD6Fz+JRrBYL7wU4CmtwXoH+j5iLJ4UxHYHv4d198ngI7hLOYH98AFcWaC5O427FTxJvM3AyJuF0+yrwQ/ha4n0N/o6D8TY8p8vcn8bvsUHI7Nk4Bq8UxniMMLRBeKtLTxjTc8UiKOJJXFoom4clJbSEY12E6/HWQl1d22sir7r20kSXbfXk9V/clDr6aKH8slT+9aqGBSxJ9BvF4skwWXiRUeEh88gm8TNMK+lzcknZQswUTqSl3s43AT/HX/FF5TvfdOGlNwujKI6dzTGPRVhsrFecLpQ3ireX8LNfBZ+fT22+OiBvTeZCGFO7gre6uC2Nc3JJXV3bqysv6ttLE122NZDXy1Jnm0oGmyZW7Q7hYXrhqtTXmSV1c1Ld73JlE4Xid+CFtbjuoKXe4jsbe/EGjChffK9O5ddW9PGEOFL3iwtSfytqtJmr45QG4a3pXNrGZ/Flen/I2LBjPG2vSl5FtDQLUzJU6bKtT3nlJ7oofW8WRpnHdqzDVLGd98L09C3zpFnZ0WJ7hmNxGG7AYyIu+YRYIFWxyCCYhYuwHGu70N0vjjfz7buDE4t2mtg9+8W/03d3jTaL0/fPA/I2yFyeJY5mFwidLNQ9bi/DB9P3m8bGfONpe1XyGm9002VteWVHr3Mq6q9I9R/ug7HvJ9qPlNRlHnBUR5gfS/+vEEIrBqq/0ntHbOnPk03CnSKOmJLKRlQnXJYKg9iClbgQP8S/hLEUj3Ddxr0rjfOmLnTnJn6+LBIDo/iT8vnX5a3JXNrKkwcbsaDLPPKYIpzqHhFnFzGI7dWRVx4tzXe+brpsayCvlbpn/LKz9Pl9MHdqon0ABxaYvibH0Imp/ML0f7fw0G8UCZDZuDHV/bLHmC39CfMzwgjyO+qI7nNfIrJpeWHeL+bZLy5N7a7vQbe5MM5PRZBfhbq81aVfJnamg8XuM0fEX3tFBntuj/nAGWmc1RX1g9heXXllaGm++LrpspG8egngC6n+k30wN1EcIbP06kqR/Vkv0rX3pboTEv0lOinoInNTRNZqVPcjaEtvYc4XC/ySQvmI6rmfl9pcJmKTqeLInCUIin2V4axEe499nVE3HIxTxA79cBpzUN7GYy4ZMgNc1QftukS7uKJ+PGyvH3nl0dJs8TXRJT3kNZ7HTmKXO0ekd3eKgP5GvEIn6zUv0Z6v44HLcGWqP7vLeC3dhTlJKGaDOJPnMaJc+VmfPy7pb6pIHuwRhlyFM1Mfd+vEwnVwqLjzWj8gb3Xpe2FG6q/X1dORie5B1XHPeNpelbyKaKm/+AbR5Rh55RMu96bvyysaz0zf+/ocbDe+JBbYFByANwvjnycW5N2FsbdV9PVY+k6pqO8H+4u5zRIxTv6osizRfCP9z+5osruoNSX9PYU7hAyPqhhzqTCc9SLoLruU7oW/CZnNtm+ipC5vg86liC3p2ysD2S3RkmE8ba9KXoNiUF2OkdekXGWmlBOEEvJZp2niknOneLEyCE4TdzTf1ckYrRWLdSaeKbJyecxJ3/YA4+4SBlCGo4XR3SoM4bZUnu2QVcF7Vl7kl8jWXiR2/uPFhXZTvCh988Zbl7dB5lKGLAQoy2hn2E/oe69q2TP+tlcmr0EwHrrsKa+6F52Hi1cCZRfgB5SUvUoE+9uNPd5cncb4XKH8eKGMbTpXE2VoaR5Ajyg/dr5LJ259caHuxMTXTmOfM306tbtTf3HBEcqPMRN1kg3rBuStyVxmV/B/qAgRRkU6vQrZ073rutBkqGN7TeRVREt/9lJHl7Xk1et52T3icnah2PKPte8Zv506PszYXek3QpnrxWKbLd5n7hLPc24q0B8kBDZDpIzvSH2fkpg+VbxBzGOJzlOm6SLtuzG1JzzUuXpjRBw9i8/LJiY+j0tzWCWMd5Y4xk0Qx5HluTZn4DvC664QbwGLaCeaDEtF3LNWvLjZKhIIC4ST2iwywBsG4K3JXEZEkmONuADfLmzkJLGr3SD0U7Vb3oLXiRctvRZgHdtrIi/q20tdXY4YTF4OwbfFe7anxRl6ufIV3RYL46UldR8Xr1i2iQW3SXivMtoMBwpPtymNvVUooupydUT5nUr2a3cZq6yfsmzbZKHs20XSaLc4v6/WydbW4ans2mQOviKONY+mMR7Hb1N/VR63Lm916RfgB+Kd5TYRJvxDvB453VjnnccsvRMtRfRre03lNaKevfSiL+pyEHkNMcQQQwwxxBBDDDHE/wL/AcjRWzfIjqPpAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle 0.9614823885703125$"
      ],
      "text/plain": [
       "0.9614823885703125"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1-0.55**9-9*0.45*0.55**8"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 0.9614823885703125$"
      ],
      "text/plain": [
       "0.9614823885703125"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1-prob_densB(1)-prob_densB(0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "def p(x):\n",
    "    return prob_densB(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 0.2127570069023438$"
      ],
      "text/plain": [
       "0.2127570069023438"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import sympy as sp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "p = sp.symbols(\"p\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}p + 1 & \\frac{1}{p}\\\\p^{2} - 1 & \\left|{p}\\right|\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[   p + 1,    1/p],\n",
       "[p**2 - 1, Abs(p)]])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = sp.Matrix([[p+1, 1/p], [p**2-1, sp.Abs(p)]])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "[1, 1]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = sp.Array([1, 1])\n",
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "ename": "AttributeError",
     "evalue": "'list' object has no attribute 'rows'",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mAttributeError\u001b[0m                            Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-5-57fc85bd3ce3>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[0mx\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mA\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0msolve\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;36m1\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;36m1\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      2\u001b[0m \u001b[0mx\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32mC:\\ProgramData\\Anaconda3\\lib\\site-packages\\sympy\\matrices\\matrices.py\u001b[0m in \u001b[0;36msolve\u001b[1;34m(self, rhs, method)\u001b[0m\n\u001b[0;32m   4500\u001b[0m         \u001b[1;32mif\u001b[0m \u001b[0mmethod\u001b[0m \u001b[1;33m==\u001b[0m \u001b[1;34m'GJ'\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   4501\u001b[0m             \u001b[1;32mtry\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 4502\u001b[1;33m                 \u001b[0msoln\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mparam\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mgauss_jordan_solve\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mrhs\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   4503\u001b[0m                 \u001b[1;32mif\u001b[0m \u001b[0mparam\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   4504\u001b[0m                     raise ValueError(\"Matrix det == 0; not invertible. \"\n",
      "\u001b[1;32mC:\\ProgramData\\Anaconda3\\lib\\site-packages\\sympy\\matrices\\matrices.py\u001b[0m in \u001b[0;36mgauss_jordan_solve\u001b[1;34m(self, B, freevar)\u001b[0m\n\u001b[0;32m   2971\u001b[0m         \u001b[1;32mfrom\u001b[0m \u001b[0msympy\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mmatrices\u001b[0m \u001b[1;32mimport\u001b[0m \u001b[0mMatrix\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mzeros\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   2972\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 2973\u001b[1;33m         \u001b[0maug\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mhstack\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mcopy\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mB\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mcopy\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   2974\u001b[0m         \u001b[0mB_cols\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mB\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mcols\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   2975\u001b[0m         \u001b[0mrow\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mcol\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0maug\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;33m:\u001b[0m\u001b[1;33m-\u001b[0m\u001b[0mB_cols\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mshape\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32mC:\\ProgramData\\Anaconda3\\lib\\site-packages\\sympy\\matrices\\common.py\u001b[0m in \u001b[0;36mhstack\u001b[1;34m(cls, *args)\u001b[0m\n\u001b[0;32m    379\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    380\u001b[0m         \u001b[0mkls\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mtype\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0margs\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 381\u001b[1;33m         \u001b[1;32mreturn\u001b[0m \u001b[0mreduce\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mkls\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mrow_join\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0margs\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m    382\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    383\u001b[0m     \u001b[1;32mdef\u001b[0m \u001b[0mreshape\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mself\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mrows\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mcols\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32mC:\\ProgramData\\Anaconda3\\lib\\site-packages\\sympy\\matrices\\common.py\u001b[0m in \u001b[0;36mrow_join\u001b[1;34m(self, other)\u001b[0m\n\u001b[0;32m    481\u001b[0m             \u001b[1;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_new\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mother\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mrows\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;36m0\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;33m[\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mrow_join\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mother\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    482\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 483\u001b[1;33m         \u001b[1;32mif\u001b[0m \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mrows\u001b[0m \u001b[1;33m!=\u001b[0m \u001b[0mother\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mrows\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m    484\u001b[0m             raise ShapeError(\n\u001b[0;32m    485\u001b[0m                 \"`self` and `rhs` must have the same number of rows.\")\n",
      "\u001b[1;31mAttributeError\u001b[0m: 'list' object has no attribute 'rows'"
     ]
    }
   ],
   "source": [
    "x = A.solve([1, 1])\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "p=0.0000000000001"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": [
    "A = np.matrix([[p+1, 1/p], [p**2-1, np.abs(p)]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "b=np.array([1, 1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[-1.e+00,  2.e-13]])"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = np.linalg.inv(A).dot(b)\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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