Data_Analysis_HW1.ipynb

{
 "cells": [
  {
   "attachments": {
    "1d_randomwalk.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![1d_randomwalk.png](attachment:1d_randomwalk.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Random Walk\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As a physics student sometimes you see some random movement in nature like brownian motion(it's actually deterministic). we can use stochastic litrature to study brownian motion and find lots of valuable information about this process. Random walk is somehow one of the most instructive models that can help you understand how random process can model the nature."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "for the beging let's see what random walk is. according to Wikipedia(https://en.wikipedia.org/wiki/Random_walk) random walk is a mathematical object, known as a stochastic or random process, that describes a path that consists of a succession of random steps on some mathematical space such as the integers.\n",
    "\n",
    "An elementary example of a random walk is the random walk on the integer number line, ${\\displaystyle \\mathbb {Z} }$ , which starts at 0 and at each step moves +1 or −1 with equal probability. Other examples include the path traced by a molecule as it travels in a liquid or a gas (see Brownian motion), the search path of a foraging animal, the price of a fluctuating stock and the financial status of a gambler: all can be approximated by random walk models, even though they may not be truly random in reality."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We know that you are passionate and want to explore all details of these process(joke :) ) but for the begining let's start with the simplest model, one dimensional random walk with same probability for right and left step."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. the simplest model of random walk is a One-dimensional random walk, which starts at 0 and at each step moves +1 or −1 with equal probability. consider 10000_step 1d random walk. and use $10^5$ ensemble. in each ensemble begin from the origin(0) and record the endpoint at 10000th step. plot the PDF of these endpoints. also, calculate the first four moments of this PDF."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#code here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.stats import norm, skew, kurtosis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "ens = 100000\n",
    "step = 10000\n",
    "nbin = 40"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "final_step = np.zeros(ens)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in range(ens):\n",
    "    ranPt = np.array(np.random.random((1, step)))\n",
    "    a = len(np.where(ranPt >= 1/2)[0])\n",
    "    b = step - a \n",
    "    final_step[i] = a - b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "maxDis = max(final_step)\n",
    "minDis = min(final_step)\n",
    "binS = np.linspace(minDis, maxDis, nbin)\n",
    "\n",
    "pdf_final, binsPdf = np.histogram(final_step, binS, density = True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mean:  0.1725\n",
      "variance:  9933.566763750001\n",
      "skewness:  -0.013871061435354156\n",
      "kurtosis:  -0.01798874208764145\n"
     ]
    }
   ],
   "source": [
    "mean_final =  np.mean(final_step)\n",
    "print('mean: ', mean_final)\n",
    "\n",
    "var_final = np.var(final_step)\n",
    "print('variance: ', var_final)\n",
    "\n",
    "skew_final = skew(final_step)\n",
    "print('skewness: ', skew_final)\n",
    "\n",
    "kurt_final = kurtosis(final_step)\n",
    "print('kurtosis: ', kurt_final)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "now look at the plot you create. does it looklike a normal distribution? if yes isn't it strange cause we know that the probability is bionomial not guassian.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 360,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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00UeF39T37t1LQEAA4eHhJabTHjp0KM8991zh66If9O+//z5ZWVkcO3aMVatW0bt37/OKPy0tjcmTJzN16tQy799QnRwlBxFJEpE9IpIiItNKWS8i8oxn/TYRia+oroj81VM2WUSWiUgbz/JIEcn0LE8WkZnF388Yb5ozZy1nzmRz++2X1fh7i7jPHnbvDWL79tQaf39fEh4ezm233Ub37t25+uqrHX1Iv/baa3Tp0oW4uDhuuOEG5syZQ0BAAKNGjWLBggWFHdLPPPMMmzZtIjY2lujoaGbO/PljKjExkZEjR9K3b1/+/Oc/l9nfUJqC5qZu3bpx5ZVXMnTo0LPuHV2TKpyyW0QCgK+BIUAqsBEYr6q7ipQZAfwO932k+wBPq2qf8uqKSGNVPempfycQraqTRSQS+FBVY5zuhE3ZbWpKbq6Lvn0foWPHFrz11tl3xk1Nrpmb85w8JYy8oRVXDcng3illd0xX1ayt/jxld1V76KGHzuq4rk2qY8ruRCBFVfd7NjYPGAPsKlJmDPCq517S60QkXERaA5Fl1S1IDB4NAN+/sYTxa6nJ41iyMoTDh5vyp9v3k5r8uVfiaNxIGTwwkyWfuDumQ0PsX8dUPSfJoS3wfZHXqbjPDioq07aiuiLyKHAjkA5cXqRclIhsBU4C/6Oq3vkvNKYIVZjzbkMi2+dxSUK2V2O5JimDRSvcHdOjh2Z6NRbzs4Irqv2Bkz6H0npCin9VKatMuXVV9X5VbQ/MAaZ6Fh8GLlDVnsDdwFwRKTGpjIhMEpFNIrIpLS3NwW4YUzmbtwWxZ18g468+TT0vD+WIjc4lqn0u79VQx7Q/3DGyLjuf35+TP/FUoH2R1+2AQw7LOKkLMBe4DkBVs1X1mOf5ZmAfUOLGrao6S1UTVDUhIiLCwW4YUzlz3m1A0yYuhl/h/W/qBR3TO/YEsfeb6h2RHhISwrFjxyxB+ChV5dixY4SEnNv1OE7+qjYCnUQkCjgIjAMmFCuzEJjq6VPoA6Sr6mERSSurroh0UtW9nvqjga88yyOAn1TVJSIdgU7A/nPaK2Oq2N69R1izMYT/mnCKkOCKy9eE4VdkMuM/7ium//Tb6rtiul27dqSmpmJn6L4rJCSk8CJBpypMDqqaJyJTgaVAAPCSqu4Ukcme9TOBRbhHKqUAGcBvyqvr2fRjItIFyAe+BSZ7lg8CHhaRPMAFTFbVn85pr4ypYrNnf0pQoPLLqzK8HUqh8MbKFQOyWPxJKL/7zUnO8YuhY4GBgURFRVXPxk2t5eh8VFUX4U4ARZfNLPJcgSnF65VV17P8ujLKvwO84yQuY2rCsWOnefvtTYwYnEmz8Hxvh3OWq5MyWLIylOWrQ7nqSu83dxn/YVdIG1OBl19eTVZWLuOvrrr7Q1eVnjE5dGiXV2Md06busORgTDmysnJ5+eU1DB7clagL8rwdTgki7rOHbbuD2PetTZVmqo4lB2PK8c47mzl27DS33355xYW9ZOTgDALrK+8vsbMHU3UsORhThvz8fGbNWkW3bm3p3/8ib4dTpvAmyuX9s1i0IpQs716bZ/yIJQdjyrBy5Vfs3XuEyZMv88qsmOfi6qQMTp6uxyerS04hbcz5sORgTBn+/e9VtG7dhNGje3o7lAr1is2hzS/y+PizmrnxkPF/lhyMKcWOHQdZvXovt9wykMDA6r0/dFUQgUv7ZrFhazBnMmr3WY7xDZYcjCnFrFmrCAsLYuLEft4OxbFL+2WTmyes21JLLuE2Ps2SgzHFHD58gvfe28L48X0ID/edEUCx0TmEN3bx6VprWjKVZ8nBmGJeeulz8vOV22671NuhnJP6ATAgMZs1G4PJq32XZBgfY8nBmCLOnMnm9dfXMmJELBdc0Nzb4ZyzS/tlcep0PbbsCPJ2KMbHWXIwpoh589aTnp7plftDV4U+PbMJDlZrWjKVZsnBmCLmzdtAz54X0KtXpLdDOS8hIe4E8dm6ELv/gqkUSw7GeHz33TF27jzIqFFx3g6lUi7tm8WRtAC2b0/1dijGh1lyMMZj8eLtAAwf3t3LkVTOgD7Z1KunLF26w9uhGB9m0zga47Fo0Taio9vQoUMLb4dSKU2b5BPbNYcP3vuI8UNeKbVMu7h5NRyV8TV25mAMcPToSTZtOsCIEbHeDqVKXNovm5RvAjn4Q+2/utvUTo6Sg4gkicgeEUkRkWmlrBcRecazfpuIxFdUV0T+6imbLCLLRKRNkXXTPeX3iMiwyu6kMRVZunQHqurzTUoFLu2XBcBn6+xqaXN+KmxWEpEAYAYwBEgFNorIQlXdVaTYcKCT59EHeAHoU0HdJ1T1z573uBN4AJgsItHAOKAb0AZYLiKdVdVVJXtsTCkWL95OVFQLGmbdTWqyt6OpvHatXVzYIZdP14Yw/urac99r4zucnDkkAimqul9Vc4B5wJhiZcYAr6rbOiBcRFqXV1dVTxap3wDQItuap6rZqvoNkOLZjjHVIj09k9WrvyYpqTu1fGbuc3JpvyySdwZx4qQf7ZSpMU6SQ1vg+yKvUz3LnJQpt66IPCoi3wMTcZ85OH0/Y6rMihW7yMvLZ/hw/+hvKDCobzb5+cLqDXZBnDl3TpJDaV87il9dU1aZcuuq6v2q2h6YA0w9h/dDRCaJyCYR2ZSWllZq4MY4sWjRNlq1akx8/AXeDqVKde2US8vmNhGfOT9OkkMq0L7I63bAIYdlnNQFmAtcdw7vh6rOUtUEVU2IiIhwsBvGlJSRkcPKlV+RlNSdevX8a/CeCAzqm8W6LcFkZXk7GuNrnPw3bAQ6iUiUiATh7ixeWKzMQuBGz6ilvkC6qh4ur66IdCpSfzTwVZFtjRORYBGJwt3JveE898+Ycn366R4yM3P8ZpRScZf2yyI7W9iQbKOWzLmpcLSSquaJyFRgKRAAvKSqO0Vksmf9TGARMAJ353EG8Jvy6no2/ZiIdAHygW+Bgu3tFJH5wC4gD5hiI5VMdVmyZBvh4WH063eRt0OpFvHdc2gQls+n60IY1Dfb2+EYH+LoCmlVXYQ7ARRdNrPIcwWmOK3rWX5dKcUL1j0KPOokNmPOV26ui2XLdjJ0aDefuBXo+QgMhP69s1m9PhiXCwL8czdNNfCvRlZjzsHatSmkp2eSlORfo5SKu7RvFsfTA9j+VaC3QzE+xJKDqbMWL95OaGgQl17axduhVKt+vbOpX9/u8WDOjSUHUyfl5+ezZMl2Lr/8YsLC/PuuaQ3DlN49svl0bQh2iwfjlCUHUydt2fIdR46c9JuJ9ioyqG82qYfrs/87m4jZOGPJwdRJixdvo379egweHO3tUGrEwD7uCx3WbrIhrcYZSw6mzlFVFi/ezoABnWnSJNTb4dSIli3yad8mj607/LsJzVQdSw6mzvnqq8McOPCj3174VpaeMTkk7wwiP9/bkRhfYMnB1DmLF29HRBg2LMbbodSo+O45nDpdj5QD1u9gKmbJwdQ5ixZtIyEhkpYtG3s7lBrVMyYHwJqWjCOWHEyd8u23P7Jr16E6M0qpqNatXLRumcfW7ZYcTMUsOZg6ZenSHQAkJdWtJqUCcTE5bN0RhNoFD6YClhxMnbJ69V46doygQ4cW3g7FK+K753A8PYCUlKPeDsXUcpYcTJ3hcuWzfv1+LrnEP2dgdaKg32Ht2hQvR2JqOxu2YOqMnTsPcupUlt9Oz+1E+zYuWjRz8cni17gi9tlSy7SLm1fDUZnayJKDqTO++ML9bTmy8ZOkJtfNwf4i7qYld7+D+7UxpbFmJVNnfPFFCh07RhDRvG4mhgJxMTmkHQvg4A92cwdTNksOpk6w/oafxXd39ztssSGtphyOkoOIJInIHhFJEZFppawXEXnGs36biMRXVFdEnhCRrzzlF4hIuGd5pIhkikiy5zGz+PsZc66sv+FnUe3zCG/ssovhTLkqTA4iEgDMAIYD0cB4ESk+leVwoJPnMQl4wUHdj4EYVY0FvgamF9nePlWN8zwmn+/OGVOgoL+hX78LvRyJ94m4Ry3ZmYMpj5Mzh0QgRVX3q2oOMA8YU6zMGOBVdVsHhItI6/LqquoyVc3z1F8HtKuC/TGmVAX9Db/4RRNvh1IrxHfP4fCR+vxw1FqWTemc/GW0Bb4v8jrVs8xJGSd1AW4BFhd5HSUiW0XkUxEZ6CBGY8pk/Q0l/TzPkt3fwZTOSXIobbBb8WvvyypTYV0RuR/IA+Z4Fh0GLlDVnsDdwFwRKTFDmohMEpFNIrIpLS2tgl0wdZn1N5R0YWQejRrmW7+DKZOT5JAKtC/yuh1wyGGZcuuKyE3AVcBE9Uz2oqrZqnrM83wzsA/oXDwoVZ2lqgmqmhAREeFgN0xdZf0NJQUEQI9o63cwZXOSHDYCnUQkSkSCgHHAwmJlFgI3ekYt9QXSVfVweXVFJAm4FxitqhkFGxKRCE9HNiLSEXcn9/5K7aWp06y/oXTx3XP47mB9fvzJ+h1MSRX+VXg6jacCS4HdwHxV3Skik0WkYCTRItwf4CnAbOCO8up66jwHNAI+LjZkdRCwTUS+BN4GJqvqT5XfVVMXWX9D2ez+DqY8jqbPUNVFuBNA0WUzizxXYIrTup7lpf63quo7wDtO4jKmItbfULYuF+USFppP8o4ghgzK8nY4ppax80nj16y/oWz1AyDW+h1MGSw5GL9m/Q3l6xmTw75vAzlx0mbgM2ez5GD8lvU3VCze0++QbP0OphhLDsZvWX9Dxbp2ziU4SNliycEUY8nB+C3rb6hYUCDEXJxDsl0pbYqx5GD8lvU3ONMzJoev99fn9BnrdzA/s+Rg/JL1NzgX3z2H/Hzhy13WtGR+ZsnB+CXrb3Au5uIc6tdXG9JqzmLJwfgl629wLiQYunXOtSulzVksORi/ZP0N56Zn9xx27w0kM8v6HYybJQfjd6y/4dwlxGbjcgmbt9nZg3Gz5GD8jvU3nLueMTmEheazeoMNaTVulhyM37H+hnMXGAh947NZvSEEz61VTB3naFZWY3yJ9TecnwGJ2XyyJpSV795A5wvzSi3TLm5eDUdlvMXOHIxfsf6G89cvIRsR5fMNId4OxdQClhyMX9mxI9X6G85T86b5dOucyxrrdzBYs5LxM8uX70JE6Bj+JKnJ+d4Ox+f0T8xm1usNOXa8Hs2b2vGry+zMwfiVZct20qtXB/tgO08DE7NQFdZusrOHus5RchCRJBHZIyIpIjKtlPUiIs941m8TkfiK6orIEyLylaf8AhEJL7Juuqf8HhEZVsl9NHXEwYPH2b49lWHDYrwdis/q1DGPls1dNqTVVJwcRCQAmAEMB6KB8SISXazYcKCT5zEJeMFB3Y+BGFWNBb4GpnvqRAPjgG5AEvC8ZzvGlGvZsp0ADBvW3cuR+C4R6J+YxbotweTmejsa401OzhwSgRRV3a+qOcA8YEyxMmOAV9VtHRAuIq3Lq6uqy1S1YLzcOqBdkW3NU9VsVf0GSPFsx5hyLV26nY4dI7joopbeDsWnDUjMJiOzns21VMc5SQ5tge+LvE71LHNSxkldgFuAxefwfojIJBHZJCKb0tLSHOyG8WcnT2aydu0+a1KqAr17ZBMcpKy2Ia11mpPkUNpMXMUvoSyrTIV1ReR+IA+Ycw7vh6rOUtUEVU2IiIgopYqpSz75ZDe5uS5LDlUgJAQSemTz+fpg7GLpustJckgF2hd53Q445LBMuXVF5CbgKmCi/nzNvpP3M+Ysy5btpHnzhvTqFentUPzCgMRsDv5QnwPf22j3uspJctgIdBKRKBEJwt1ZvLBYmYXAjZ5RS32BdFU9XF5dEUkC7gVGq2pGsW2NE5FgEYnC3cm9oRL7aPxcbq6LTz7ZxZAh3QgIsNHZVWFAYhaAjVqqwyr8T/J0Gk8FlgK7gfmqulNEJovIZE+xRcB+3J3Hs4E7yqvrqfMc0Aj4WESSRWSmp85OYD6wC1gCTFFVV1XsrPFPa9emcPJkFkOHdvN2KH6jVUQ+nTrmWnKowxydM6rqItwJoOiymUWeKzDFaV3P8jLnN1DVR4FHncRmzNKlOwgJCWTQoC7eDsWvDOidxatvNST9lNCkkXU+1DV2Dm58mqqydOkOBg3qQliYDb2sSgP6ZOPKF9ZttrOHusiSg/FpO3ce4tChEzZKqRpEd8qlaROXDWmtoyw5GJ+2dOl2RIQhQ4pftG8qKyAALumdzdpNweRZr1+dY8nB+LSlS3fQq1cHWrRo5O1Q/NKAxGxOnq7H9t2B3g7F1DBLDsZnpaYeZ8eOg9akVI36xGcTEGBXS9dFlhyMz1q2bAdgE+1Vp4ZhSnz3HBvSWgdZcjA+a+nSHVx4YUubaK+aDUjM4pvvAjl42CZHrkssORif5J5oL8WalGrAgMRswK6WrmssORif9Mknu8nLy7eromtA+zYuItvnWb9DHWPJwfikpUt32ER7NWhAYhabtweRnp7p7VBMDbEpF43PycnJ45NPdjNyZA+baK+GXN4/i9ffaci8F+9g5ODSE0S7uHk1HJWpTpYcjM9Zu3Yfp05lkdB5ManJ73k7nDohpksurSJcrPg8pMzkYPyLfe0yPmfZMvdEe4lx2d4Opc4QgSv6Z7J+SzCnz5R2Py7jbyw5GJ9SdKK9EOsfrVGDB2aRmyd8ts4OfF1gycH4lM2bv+XQoRMMH24XvtW0mC65tGzhYsVqSw51gSUH41Pmz99AaGgQI0fGejuUOqdePbhiQCbrNlvTUl1gycH4jMzMHBYu3MrIkbE0bGjfXr3hygHupqXP19vx93eOkoOIJInIHhFJEZFppawXEXnGs36biMRXVFdExorIThHJF5GEIssjRSTTc+vQwtuHGrN06Q5Onsxi7Nje3g6lzoq5OJeWza1pqS6oMDmISAAwAxgORAPjRaT45PnDgU6exyTgBQd1dwDXAp+V8rb7VDXO85hcynpTB7311kbatAmnf/8y7zBrqpm7aSnL3bSUYU1L/szJmUMikKKq+1U1B5gHjClWZgzwqrqtA8JFpHV5dVV1t6ruqbI9MX7t8OETfPrpHsaO7U29etYa6k2DB2SSkyusXm9zLfkzJ/9lbYHvi7xO9SxzUsZJ3dJEichWEflURAaWVkBEJonIJhHZlJaW5mCTxpe9885m8vPVmpRqge5d3U1Lyz8P9XYopho5SQ6lnTuqwzJO6hZ3GLhAVXsCdwNzRaRxiY2ozlLVBFVNiIiIqGCTxpepKm+9tZHevaPo2NF+195Wr557Og1rWvJvTpJDKtC+yOt2wCGHZZzUPYuqZqvqMc/zzcA+oLODOI2fSk7+jr17j/CrX9lZQ21x5UBrWvJ3TpLDRqCTiESJSBAwDlhYrMxC4EbPqKW+QLqqHnZY9ywiEuHpyEZEOuLu5N5/Tntl/Mr8+RsJCQlk1Kg4b4diPAqallastqYlf1VhclDVPGAqsBTYDcxX1Z0iMllECkYSLcL9AZ4CzAbuKK8ugIhcIyKpQD/gIxFZ6tnWIGCbiHwJvA1MVtWfqmRvjc/Jysrl/fe3kpTUncaN7YOotihoWlq7KZgz1rTklxzNyqqqi3AngKLLZhZ5rsAUp3U9yxcAC0pZ/g7wjpO4jP/7+OOdnDiRYU1KtdDggZm8ubABqzcEM+yyLG+HY6qYjQk0tdpbb22kdesmDBxo3U61TWzXXCKsaclvWXIwtdbRoydZufIrrrsuwW7qUwsVbVrKyLSmJX9j/3Gm1nr33c24XPl2bUMtNnhAJtk5wuoNNmrJ31hyMLWSqjJ//kbi4zvQqVMrb4djytAjOpcWzdx3iDP+xZKDqZW2b0/lq68OW0d0LVfQtPTFphDOnLE78/kTSw6mVpo/fyPBwfUZPbqnt0MxFRg8IIvsHGH58l3eDsVUIUdDWY2pSTk5eSxYsIWhQ2MIDw/zdjimAj2ic2je1MWbr82iV4fjZZZrFzevBqMylWVnDqbWWbFiF8ePn+H66xO9HYpxICAAki7P5PP1wfz4k32k+Av7TZpa5803N9KyZSMGDbJrG3zF1UkZuFzCBx/bNQ/+wpqVTK1y9OhJVqzYxaRJl/LDjoneDsc41KGdi95x2SxYFMaNvzxDQIC3IzKVZWcOplaZP38jLlc+48f39XYo5hxdOyKDH9Lqs3azXfPgDyw5mFpDVZk3bz19+nTkootaejscc44u7ZtF86Yu3l1kgwj8gSUHU2usX7+f/fvTmDDBzhp8Uf36MGZYBms2BnP4iLUr+TpLDqbWmDt3HY0ahTByZA9vh2LO09VJGYjAe0usY9rXWXIwtUJ6eiYffvgl11wTT1hYkLfDMefpFy3z6d87m/eXhZGX5+1oTGVYcjC1woIFm8nKyrUmJT9w7YgMfjoewKq1Nt+SL7PkYGqFuXPX0a1bW7p3b+ftUEwl9Y3PpnWrPOuY9nGOkoOIJInIHhFJEZFppawXEXnGs36biMRXVFdExorIThHJF5GEYtub7im/R0SGVWYHTe23bdv37NhxkAkT+iJi9wXwdQEBcE1SBpu+DObbVOuY9lUVJgcRCQBmAMOBaGC8iEQXKzYc6OR5TAJecFB3B3At8Fmx94sGxgHdgCTgec92jJ964431hIQEcs018RUXNj5h1NBM6tdXO3vwYU7OHBKBFFXdr6o5wDxgTLEyY4BX1W0dEC4ircurq6q7VXVPKe83Bpinqtmq+g2Q4tmO8UMZGTksWLCZESNibZI9P9K8aT6XX5LFR8vDyLKZvH2Sk+TQFvi+yOtUzzInZZzUPZ/3Q0QmicgmEdmUlpZWwSZNbbVo0ZecPJllHdF+6NoRGZw8Xc/uMe2jnCSH0hqB1WEZJ3XP5/1Q1VmqmqCqCRERERVs0tRWc+euJyqqBf36XejtUEwVi++eQ2T7PN75yM4IfZGT5JAKtC/yuh1wyGEZJ3XP5/2MH9i37yjr1u1j3Lg+1hHth0Tg2uFn2PFVEF/vszk+fY2T5LAR6CQiUSIShLuzeGGxMguBGz2jlvoC6ap62GHd4hYC40QkWESicHdybziHfTI+Yt689QQE1LNbgfqxEYMzCQ5S3l1sZw++psJ0rqp5IjIVWAoEAC+p6k4RmexZPxNYBIzA3XmcAfymvLoAInIN8CwQAXwkIsmqOsyz7fnALiAPmKKqrirda+N1ubku5s/fyODB0bRq1cTb4Zhq0riRMmRQJktWhrLni/E0CCu9VdnuElf7ODrXU9VFuBNA0WUzizxXYIrTup7lC4AFZdR5FHjUSWzGNy1fvou0tFMM6/cZqckfezscU42uG5nBh8vDWLQilLGjMrwdjnHIrpA2XvHGG+to1aox/RJsnKO/i+6cS/euObw8vyGZWda35CssOZgad/jwCT75ZDfXX59Ifbu80e+JwO9uOUnasQDeeK+Bt8MxDllyMDXuzTc3kJ+vjBvXx9uhmBoS1y2XS/tl8dpbDfjphH3s+AL7LZkadfJkJrNnf8all3YhMrKFt8MxNWjqb06SlS3839yG3g7FOGDJwdSoF15YyfHjZ5g+faS3QzE1rEM7F1cPz+DdxWE2IZ8PsORgaswPP6Tz73+vYsyYnsTGtq+4gvE7/zXhNMFByguvNPJ2KKYClhxMjfnf/12Cy5XPtGkjvB2K8ZLmTfO54bozfLImlG27Ar0djimHJQdTI/buPcIbb6znxhsvoUMH62uoyyZce4bmTV0883+N0YpmWjNeY8nB1Ii///0jwsKCuOuuod4OxXhZaIhy+w2n2LY7iFVf2K1EaytLDqbabdiwnyVLtjNlymCaN7eRKgauGpJJ1AW5PPefRuTleTsaUxpLDqZaqSqPPPIBrVo15rbbBnk7HFNL1A+Aqb85xfeH6vPeEpuUrzay5GCq1ZIlO9i06QD33DOMsLBgb4djapEBidn0is1m9pyGnDqV5e1wTDE2ybqpNnl5Lv7+9w+56KKWdjW0KcE9rcYpbr6rBY8/OJXJN54utZzN2OodduZgqs2bb24gJeUo06dfRX2bRMmUIrpzLkMvzWTOgoYc/dE+jmoTO3Mw1SIjI5snn1xCQkIkMb94lNRkb0dkaqvf3nSKVV+E8PSLjXl02glvh2M8LFWbajF79mccOXKS//mfUdgdQE152v7Cxc3Xn+bjz0JZtyXI2+EYD0sOpsodO3aaGTNWMGxYDImJHb0djvEBN449zQVt8/jHjCZk2S0+agVHyUFEkkRkj4ikiMi0UtaLiDzjWb9NROIrqisizUTkYxHZ6/nZ1LM8UkQyRSTZ85hZ/P1M7fbXvy4kIyOH++67ytuhGB8RFAj/PSWd1MP1efUtuxamNqgwOYhIADADGA5EA+NFJLpYseFAJ89jEvCCg7rTgBWq2glY4XldYJ+qxnkek89350zN++CDZObP38iddw6hU6dW3g7H+JDEuByGXZbJK/Mb2qyttYCTM4dEIEVV96tqDjAPGFOszBjgVXVbB4SLSOsK6o4BXvE8fwW4unK7Yrzt0KET3HvvW/TseQF/+INNk2HO3e//6yTBwco/nm9i8y55mZPk0Bb4vsjrVM8yJ2XKq9tKVQ8DeH62LFIuSkS2isinIjKwtKBEZJKIbBKRTWlpaQ52w1Sn/Px87rprLjk5eTz77K8JDLRvfubctWiWzx03nWJjcjBLV9m8S97kJDmUNtakeE4vq4yTusUdBi5Q1Z7A3cBcEWlcYiOqs1Q1QVUTIiIiKtikqW6zZn3K6tV7+ctfrqFjR/t9mPN3zfAMojvn8NSLjTl12oa6eYuT5JAKFL0zSzvgkMMy5dU94ml6wvPzKICqZqvqMc/zzcA+oLOTnTHesXPnQR577COGDYthwgS7EtpUTkAATJuazon0enZTIC9ykhw2Ap1EJEpEgoBxwMJiZRYCN3pGLfUF0j1NReXVXQjc5Hl+E/A+gIhEeDqyEZGOuDu595/3HppqlZmZw9SprxMeHsaTT16P2EUNpgpcfFEeY6/K4J1FYWzd+q23w6mTKrxCWlXzRGQqsBQIAF5S1Z0iMtmzfiawCBgBpAAZwG/Kq+vZ9GPAfBG5FfgOGOtZPgh4WETyABcwWVV/qpK9NVXub3/7kD17fmDu3NttOm5TpW6/8RQrVodw951P8J+nfqS0GVhs3qXqI+oHQwISEhJ006ZN3g6jzlm16ismTPg3t9wykMm/nO/tcIwfWv55CPf9vSn33J7O9WMySqy35FA5IrJZVRNKW2dXSJvzcuzYae66ay6dO7fi/vvtYjdTPQYPyKJvryxmvtqIfd/aVHA1yZKDOWeqyp/+NJ/jxzN47rkbCA21+XBM9RCBaVNPEhqqTLmvmV0cV4MsOZhzkpfn4v7732XJku3ce+8IYmKKX/JiTNVq08rFc3/7Cc2HKdObc/CwJYiaYMnBOHbmTDa33PISL7+8mttvv4zJky/zdkimjuh4QR7P/e0nsnKEO+5rxg9H7aOrutkRNo4cOZLOtdc+xyef7OZvf7uOBx8cQ7169udjak6nqDyefeQYJ0/VY8p9zUk7Zn9/1cmOrqnQV18dZuTIp9i37yj/+c+t3HzzAG+HZOqorp3yePqvP3HsuDtB/PjjKW+H5Les+9+U67PP9nDbbS8TFhbEu+9OJTa2fcWVjKlGsV1z+edDx/n9A824dvR0nn/sGOGNSw7Jt2GulWPJwZTpjTfWc++987noola89tptaNpv7XafplaI757D/z7wE3c/1Iw7/6cZM/72E40a+v41W7WJNSuZErKz8/j73z/innvmccklF/Hee7+jbdum3g7LmLMk9szhsfuPk3IgkDumN2f3XvuuW5UsOZhCLlc+b765gQED/sazzy5n/Pg+vPbaJBo3DvV2aMaUakBiNo/ff5y0Y/W4+a4WPPJUE346YR9rVcFSrUFVWbJkB48//hFff32EHj3a87//ez2DBnXxdmjGVGhgn2zenp3Gi3Mb8ubCBqxYHcJ/TTjNH6LzCAqyj7jzZSm2jluzZi+jRj3Nrbe+hMulzJ59M4sW/cESg/EpDRsod912ijeeTyO2aw5Pv9iYwYOf4JNPdns7NJ9labUOUlW2bv2OJ55YzKef7qF163CefPJ6fvWr3tQvbepLY3xEZHsXTz18nNUbgnlqdh6//vUs+vfOYuotp7iwQ95ZZW00U/ksOdQh6emZvPvuZubMWcuuXYdo2rQBDz44hptu6k9ISCCpyeO8HaIxlSbibmrqE5/Gm+834KU3GjL+txHEds1h9LAMrhyYRViojWyqiE3Z7edUlQ0bvmHu3LV88MGXZGXlEhPTlokT+3Httb1o1Ojn+/RacjD+6Hh6PT78OJSFy8L4NrU+YaH5DBmUxX9NnU58fIc6fYOq8qbstuTgh1SVgweP8+GHXzJ37jpSUo7SsGEw11wTz5UJC+naKa/ijRjjZ1Rh265AFi4L4+PPQsjKrkfnzq2YMKEvAwd2JioqgpCQQG+HWaMsOfi5zMwctm1LZfPmA2zefIAtW77lyJGTAPTqFcnEiX0ZPTqOsLBgOzswBjidISz/LISFS8PYscc95Xy9ekqbVi46tMsjJj6JTp1acdFFLbnoolY0a9bAyxFXj0onBxFJAp7GfavPF1X1sWLrxbN+BO7bhN6sqlvKqysizYA3gUjgAPArVT3uWTcduBX3bULvVNWl5cVXF5JDXp6LI0dO8sMP6fzwQzqHD6fzzTdpbNnyLTt3HiQvLx+AyMgWXBz1Pd0vziGhRw5RF9hZgjHl+TY1gD37AjnwfX2+Ta3PN9/X57vU+uTk/tzc1LK5i84X5tLlwlwuufJOYmLa0a5dU59vkqpUchCRAOBrYAiQCmwExqvqriJlRgC/w50c+gBPq2qf8uqKyD+An1T1MRGZBjRV1XtFJBp4A0gE2gDLgc6q6iorxppMDqqKy5WPy+X+qaqex8/rVSlcnpvrIiMjp8gju/B5ZmYOmZm5ZGbmkJWVW/jIzHT/PHMmm6NH3QkhLe0U+fln/65CQ/KJ7pxL94tz6d41h25dcmkWnl8jx8EYf+ZywQ9HAziQWp9vvqvP1/sD2bPPnTzy890JoUmTULp1a0unTq0ICgpARBAR6tU7+2dgYAANGgTTsGEwDRuGeH4Ge5aFVHgtRuPGIdV2IWp5ycHJaKVEIEVV93s2Ng8YA+wqUmYM8Kq6M806EQkXkda4zwrKqjsGuMxT/xVgFXCvZ/k8Vc0GvhGRFE8Ma53usFNffvkdo0c/gzs2Cn+5nlgLy+Xn5xcmhOpshgsOUoKDlZCCnyFK86YuEnvk06qFi4jmLiKa59OyuYuIFvmEN87Hx7+4GFMrBQRA29Yu2rZ20b93duHyrGzYdyCQPfsC2bPvDF/vS2fBtq9x5bv7NPLzQRG04LWCy1W5f9J77hnGPfckVXaXzpmT5NAW+L7I61TcZwcVlWlbQd1WqnoYQFUPi0jLIttaV8q2ziIik4BJnpenRWSPg32pCS2AH70dRC1kx6VsdmzKVuePzR//+BR//GOJxVV1XDqUtcJJcigt7RX/+lxWGSd1z+f9UNVZwKwKtlXjRGRTWadpdZkdl7LZsSmbHZvS1cRxcTJ9RipQdBL/dsAhh2XKq3vE0/SE5+fRc3g/Y4wx1chJctgIdBKRKBEJAsYBC4uVWQjcKG59gXRPk1F5dRcCN3me3wS8X2T5OBEJFpEooBOw4Tz3zxhjzHmosFlJVfNEZCqwFPdw1JdUdaeITPasnwkswj1SKQX3UNbflFfXs+nHgPkicivwHTDWU2eniMzH3WmdB0wpb6RSLVTrmrpqCTsuZbNjUzY7NqWr9uPiFxfBGWOMqVo2ZbcxxpgSLDkYY4wpwZJDJYnIH0VERaRFkWXTRSRFRPaIyLAiy3uJyHbPumfE16+9L4OIPCEiX4nINhFZICLhRdbV6WNTlIgkeY5DimeWgDpFRNqLyEoR2S0iO0Xk957lzUTkYxHZ6/nZtEidUv9+/JGIBIjIVhH50PO6Zo/Lz9M/2ONcH7iH3C4FvgVaeJZFA18CwUAUsA8I8KzbAPTDfS3HYmC4t/ehmo7LUKC+5/njwON2bEocowDP/ncEgjzHJdrbcdXwMWgNxHueN8I91U408A9gmmf5NCd/P/74AO4G5gIfel7X6HGxM4fK+Rfw35x9kV7h9B+q+g3uEVyJnms5GqvqWnX/Rl8Frq7pgGuCqi5T1YIZ/9bhvlYF7NgUVTgtjarmAAVTy9QZqnpYPRN0quopYDfu2RDG4J5SB8/Pqz3PS/37qdGga4iItANGAi8WWVyjx8WSw3kSkdHAQVX9stiq8qYSSS1lub+7BfeZANixKaqsY1EniUgk0BNYT7GpdYCiU+vUlWP2FO4vnkVn0qzR42K3CS2HiCwHflHKqvuB+3A3n5SoVsqy851KpNYq79io6vueMvfjvlZlTkG1Usr73bFxqC7uc6lEpCHwDnCXqp4sp7upThwzEbkKOKqqm0XkMidVSllW6eNiyaEcqnplactFpDvutr0vPX/I7YAtIpJI+VOJtCtluU8q69gUEJGbgKuAwZ6mIqgjx8YhmyYGEJFA3Ilhjqq+61l8RERaq3tCzro4tU5/YLTnVgghQGMReZ2aPi7e7nTxhwfumxUVdEh34+zOof383Om6EejLz52uI7wdezUdjyTcV7hHFFte549NkWNR37P/UfzcId3N23HV8DEQ3P1LTxVb/gRnd7z+o6K/H3994L6tQUGHdI0eFztzqGJa/vQfvwVeBkJxfwAuLnUjvu853H+oH3vOrNap6mQ7Nj/T8qeWqSv6AzcA20Uk2bPsPvx3ap3KqtHjYtNnGGOMKcFGKxljjCnBkoMxxpgSLDkYY4wpwZKDMcaYEiw5GGOMKcGSgzHGmBIsORhjjCnh/wHxjxFlbs558wAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "rG = np.random.normal(loc = mean_final, scale = np.sqrt(var_final), size=ens)\n",
    "\n",
    "plt.hist(rG, nbin, density = True, color = '#E3CF57', label = 'RP - generated!')\n",
    "plt.plot(binS[0:-1], pdf_final, color = '#191970', label = 'Final Step PDF')\n",
    "plt.legend()\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#explain here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# The normal distribution can be used as an approximation to the binomial distribution, \n",
    "# under certain circumstances, namely: If X ~ B(n, p) and if n is large and/or p is close to ½, \n",
    "# then X is approximately N(np, npq)\n",
    "# توزیع نرمال می تواند به عنوان تخمینی از توزیع دوجمله ای استفاده شود اگر تعداد انتخاب ها بزرگ شود."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2.change the probability of jumping to right and left in a way that $P_{left} = 2 P_{right}$ and then repeat the process of Q1 again and evaluate the PDF and first four moments and cumulants. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#code here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "final_step2 = np.zeros(ens)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in range(ens):\n",
    "    ranPt = np.array(np.random.random((1, step)))\n",
    "    a = len(np.where(ranPt >= 2/3)[0])\n",
    "    b = step - a \n",
    "    final_step2[i] = a - b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "maxDis2 = max(final_step2)\n",
    "minDis2 = min(final_step2)\n",
    "binS2 = np.linspace(minDis2, maxDis2, nbin)\n",
    "\n",
    "pdf_final2, binsPdf2 = np.histogram(final_step2, binS2, density = True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mean:  -3333.36242\n",
      "variance:  8861.8564917436\n",
      "skewness:  0.010656909585879002\n",
      "kurtosis:  -0.00916674785848004\n"
     ]
    }
   ],
   "source": [
    "mean_final2 =  np.mean(final_step2)\n",
    "print('mean: ', mean_final2)\n",
    "\n",
    "var_final2 = np.var(final_step2)\n",
    "print('variance: ', var_final2)\n",
    "\n",
    "skew_final2 = skew(final_step2)\n",
    "print('skewness: ', skew_final2)\n",
    "\n",
    "kurt_final2 = kurtosis(final_step2)\n",
    "print('kurtosis: ', kurt_final2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 362,
   "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": [
    "rG2 = np.random.normal(loc = mean_final2, scale = np.sqrt(var_final2), size=ens)\n",
    "\n",
    "plt.hist(rG2, nbin, density = True, color = '#E3CF57', label = 'RP - generated!')\n",
    "plt.plot(binS2[0:-1], pdf_final2, color = '#191970', label = 'Final Step PDF')\n",
    "plt.legend()\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "according to your result, can you tell how the shape of PDF function related to its cumulants?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "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(binS[0:-1], pdf_final, color = '#458B74', label = 'Equal probability')\n",
    "plt.plot(binS2[0:-1], pdf_final2, color = '#8B0A50', label = '$P_{left} = 2 P_{right}$')\n",
    "plt.legend(fontsize = 10)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#explain here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# کامیولنت ها - و یا ممان ها را می توان به عنوان پایه هایی در نظر گرفت که شکل یک تابع را بر اساس آن ها بسط می دهیم. بنابراین اطلاعاتی \n",
    "# که در هر کدام از آن ها نشسته است بخشی از شکل را مشخص می کند. البته بسط بر اساس این پایه ها با بسط هایی مانند بسط فوریه چند تفاوت عمده \n",
    "# دارند، که می توان به یکنوا نبودن کاهش اهمیت با جلو رفتن در مراتب بسط - ممان های بالاتر - و هم چنین عدم آگاهی از این که چه میزان اطلاعات در هر \n",
    "# بسط نشسته است اشاره کرد."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Conditional Probability\n"
   ]
  },
  {
   "attachments": {
    ".jpg": {
     "image/jpeg": 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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![.jpg](attachment:.jpg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, \"Do you want to pick door No. 2?\" Is it to your advantage to switch your choice? if Yes write a it's code to tell Us why and if your answer is No code it to prove the first group they are wrong:)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#code here - Always Switch!!!!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Chance to be winner: 67.5 %\n"
     ]
    }
   ],
   "source": [
    "win = 0 \n",
    "num = 1000\n",
    "\n",
    "for i in range(num):\n",
    "\n",
    "    flag = [False, False, False]\n",
    "    rand1 = np.random.uniform(0,1)\n",
    "    if rand1 <= 1/3:\n",
    "        flag[0] = True\n",
    "    elif rand1 > 2/3:\n",
    "        flag[2] = True\n",
    "    else:\n",
    "        flag[1] = True\n",
    "\n",
    "    choice = -1\n",
    "    rand2 = np.random.uniform(0,1)\n",
    "    if rand2 <= 1/3:\n",
    "        choice = 0\n",
    "    elif rand2 > 2/3:\n",
    "        choice = 2\n",
    "    else:\n",
    "        choice = 1\n",
    "\n",
    "    if flag[choice] == False:\n",
    "        win = win + 1\n",
    "\n",
    "winp = win/num * 100\n",
    "print('Chance to be winner:', winp, '%')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Centeral limit theorem"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Choose a non-Gaussian distribution (uniform/bionomial/poission) and generate $10^5$ random variables for each of them.                                                      \n",
    "a) plot the PDF of samples you created .                                                   \n",
    "b) create $10^4$ random subset of length {2 , 5 , 10 , 20 , 100} from the first non-Gaussian distribution and caclculate mean of each subset, then plot the PDF of these means.                                                                       \n",
    "c) what do you understand form comparing the original distribution with those five?\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 172,
   "metadata": {},
   "outputs": [],
   "source": [
    "#code here\n",
    "import random"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 173,
   "metadata": {},
   "outputs": [],
   "source": [
    "numP = 100000\n",
    "numEns = 10000\n",
    "leng = 5\n",
    "lengl = [2, 5, 10, 20, 100]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 178,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Uniform\n",
    "\n",
    "ranPtU = np.array(np.random.random((1, numP)))\n",
    "f = np.zeros((leng, numEns))\n",
    "\n",
    "l = 0 \n",
    "for i in lengl:\n",
    "    for j in range(numEns):\n",
    "        a = random.sample(range(0, 10000), i)\n",
    "        sumS = 0\n",
    "        for k in a:\n",
    "            sumS = sumS + ranPtU[0,k]\n",
    "        f[l, j] = sumS/i\n",
    "    l = l + 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 179,
   "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": [
    "for i in range(leng):\n",
    "    plt.hist(f[i, :], density = True, alpha = 1- i*0.1, label = 'Subset Length: %s' % lengl[i])\n",
    "    \n",
    "plt.hist(ranPtU[0,:], density = True, histtype='step', label = 'Uniform random')\n",
    "plt.legend(loc = 2)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 238,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Binomial\n",
    "\n",
    "n, p = 10, .5\n",
    "\n",
    "ranPtB = np.random.binomial(n, p, numP)\n",
    "bino = np.zeros((leng, numEns))\n",
    "\n",
    "l = 0 \n",
    "for i in lengl:\n",
    "    for j in range(numEns):\n",
    "        a = random.sample(range(0, 10000), i)\n",
    "        sumS = 0\n",
    "        for k in a:\n",
    "            sumS = sumS + ranPtB[k]\n",
    "        bino[l, j] = sumS/i\n",
    "    l = l + 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 242,
   "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": [
    "for i in range(leng):\n",
    "    plt.hist(bino[i, :], density = True, alpha = 1 - (i)*0.2, label = 'Subset Length: %s' % lengl[i])\n",
    "    \n",
    "plt.hist(ranPtB[:], density = True, histtype='step', lw =1.75, label = 'Binomial: n = 10 & p = 0.5')\n",
    "plt.legend(fontsize = 8)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 217,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Poisson\n",
    "\n",
    "ranPtP = np.random.poisson(5, numP)\n",
    "poi = np.zeros((leng, numEns))\n",
    "\n",
    "l = 0 \n",
    "for i in lengl:\n",
    "    for j in range(numEns):\n",
    "        a = random.sample(range(0, 10000), i)\n",
    "        sumS = 0\n",
    "        for k in a:\n",
    "            sumS = sumS + ranPtP[k]\n",
    "        poi[l, j] = sumS/i\n",
    "    l = l + 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 232,
   "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": [
    "for i in range(leng):\n",
    "    plt.hist(poi[i, :], density = True, alpha = 1 - (i)*0.2, label = 'Subset Length: %s' % lengl[i])\n",
    "    \n",
    "plt.hist(ranPtP[:], density = True, histtype='step', lw =1.75, label = 'Poisson with $\\lambda = 5$')\n",
    "plt.legend(loc = 1)\n",
    "plt.xlim(-0.7, 12)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 243,
   "metadata": {},
   "outputs": [],
   "source": [
    "# همگی به سمت نرمال پیش رفته اند، و نمایش قضیه حد مرکزی محسوب می شوند"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Poission Approximation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Use python libraries to generate bionomial and poission distributions for different N’s (remember $\\lambda =NP$). Over plot them for N = {5, 10, 20, 50, 100, 200, 1000}\n",
    "How does it behave as N increases? \n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 285,
   "metadata": {},
   "outputs": [],
   "source": [
    "#code here \n",
    "numRP = 10000\n",
    "pS = 0.5\n",
    "nT = [5, 10, 20, 50, 100, 200, 1000, 10000]\n",
    "nTl = len(nT)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 286,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x864 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x864 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x864 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x864 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x864 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x864 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x864 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 864x864 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "for i in range(nTl):\n",
    "    ranPtB2 = np.random.binomial(nT[i], pS, numRP)\n",
    "    ranPtP2 = np.random.poisson(nT[i]* pS, numRP)\n",
    "    plt.subplots(figsize=(12, 12), sharex=True)\n",
    "    plt.subplot(nTl, 1, i+1)\n",
    "    plt.hist(ranPtB2, 20, density = True, color = '#191970', label = 'Binomial - # of trials: %s' %nT[i])\n",
    "    plt.hist(ranPtP2, 20, density = True, color = '#E3CF57', alpha = 0.8, label = 'Poisson')\n",
    "    plt.legend()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# هر دو به سمت یک رفتار همگرا می شوند."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Box Muller Method"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1- generate Normal distribution from uniform distribution (Box-Muller algorithm(https://en.wikipedia.org/wiki/Box%E2%80%93Muller_transform)- do not need to derive and prove it. the topic will be included in lectures in the following week). Normal= zero mean, unit variance    \n",
    "a)Sample many points from this distribution. plot the pdf of the sample (plot hist).             \n",
    "                              \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "#code here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 306,
   "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": [
    "numRPuni = 1000000\n",
    "\n",
    "ranBM1u = np.array(np.random.random((numRPuni)))\n",
    "ranBM2u = np.array(np.random.random((numRPuni)))\n",
    "\n",
    "ranBM1 = np.sqrt(-2 * np.log(ranBM1u)) * np.cos(2 * np.pi * ranBM2u)\n",
    "ranBM2 = np.sqrt(-2 * np.log(ranBM1u)) * np.sin(2 * np.pi * ranBM2u)\n",
    "\n",
    "plt.hist(ranBM1, 30, density = True, color = '#191970', histtype='step', linestyle = '-.', lw =1.75)\n",
    "plt.hist(ranBM2, 30, density = True, color = '#8B1A1A', histtype='step', linestyle = ':', lw = 2)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b)Calculate mean of the sample. Its variance."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 309,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.278755455548647e-05 1.0005714959866445\n",
      "2.278755455548157e-05 1.000570495415149\n",
      ":)\n",
      "-0.0010019801777192503 1.0001650238279507\n",
      "-0.001001980177719267 1.0001640236629226\n"
     ]
    }
   ],
   "source": [
    "#code here\n",
    "# mean\n",
    "meanBM1 = sum(ranBM1)/numRPuni\n",
    "meanBM2 = sum(ranBM2)/numRPuni\n",
    "\n",
    "meanBM1P = np.mean(ranBM1)\n",
    "meanBM2P = np.mean(ranBM2)\n",
    "\n",
    "varBM1 = sum((ranBM1-meanBM1)**2)/ (numRPuni -1)\n",
    "varBM2 =sum((ranBM2-meanBM2)**2)/ (numRPuni -1)\n",
    "\n",
    "varBM1P = np.var(ranBM1)\n",
    "varBM2P = np.var(ranBM2)\n",
    "\n",
    "print(meanBM1, varBM1)\n",
    "print(meanBM1P, varBM1P)\n",
    "print(':)')\n",
    "print(meanBM2, varBM2)\n",
    "print(meanBM2P, varBM2P)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c)Overplot the pdf now, using this mean and variance (on the plot hist). Also use the mean and std deviation functions from python library. Do they agree?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 310,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.stats import norm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 323,
   "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": [
    "#code here\n",
    "randG = np.random.normal(loc = meanBM1, scale=np.sqrt(varBM1), size=numRPuni)\n",
    "\n",
    "x = np.linspace(norm.ppf(0.0001), norm.ppf(0.9999), 100)\n",
    "plt.plot(x, norm.pdf(x),'r-', lw=3, alpha=0.6, label='norm pdf')\n",
    "\n",
    "plt.hist(ranBM1, 30, density = True, color = '#191970', histtype='step'\n",
    "         , linestyle = '-.', lw =1.75, label = 'Box Muller')\n",
    "\n",
    "plt.hist(randG, 30, density = True, color = '#8B1A1A', alpha = 0.6, label = 'RPs - mean & var')\n",
    "\n",
    "plt.legend(loc = 2)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "d)Now that you have your Normal distribution, how can you turn it into a Gaussian distribution with  given mean and variance? For example mean=2, and std dev=4 ?   "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 356,
   "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": [
    "#code here\n",
    "\n",
    "new_mean = 2\n",
    "new_std = 4\n",
    "\n",
    "randG24 = new_std * ranBM1 + new_mean\n",
    "\n",
    "randG24P = np.random.normal(loc = new_mean, scale = new_std, size=numRPuni)\n",
    "\n",
    "plt.hist(randG24, 40, density = True, color = '#8B1A1A', label = 'RP - BM')\n",
    "plt.hist(randG24P, 40, density = True, color = '#191970', histtype='step', lw =1.5, label = 'RP-generated!')\n",
    "\n",
    "plt.legend(loc = 2)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "there are two problems left, but you can choose one of them to solve.(we highly recommend you to solve both of them)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Nice Matrix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A matrix of size n$\\times$n which include numbers of {1,2,3,...,$n^2$} is called nice, if sum of all rows and columns  are the same. write a code that get n as a input and return a n$\\times$n Nice Matrix. print out the examples of Nice Matrix for n = 4,5,6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 332,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Magic Square for n = 7\n",
      "Sum of each row or column 175 \n",
      "\n",
      "20 12  4 45 37 29 28 \n",
      "11  3 44 36 35 27 19 \n",
      " 2 43 42 34 26 18 10 \n",
      "49 41 33 25 17  9  1 \n",
      "40 32 24 16  8  7 48 \n",
      "31 23 15 14  6 47 39 \n",
      "22 21 13  5 46 38 30 \n"
     ]
    }
   ],
   "source": [
    "#code here - #GeeksForGeeks \n",
    "def generateSquare(n):\n",
    " \n",
    "    # 2-D array with all\n",
    "    # slots set to 0\n",
    "    magicSquare = [[0 for x in range(n)]\n",
    "                   for y in range(n)]\n",
    " \n",
    "    # initialize position of 1\n",
    "    i = n // 2\n",
    "    j = n - 1\n",
    " \n",
    "    # Fill the magic square\n",
    "    # by placing values\n",
    "    num = 1\n",
    "    while num <= (n * n):\n",
    "        if i == -1 and j == n:  # 3rd condition\n",
    "            j = n - 2\n",
    "            i = 0\n",
    "        else:\n",
    " \n",
    "            # next number goes out of\n",
    "            # right side of square\n",
    "            if j == n:\n",
    "                j = 0\n",
    " \n",
    "            # next number goes\n",
    "            # out of upper side\n",
    "            if i < 0:\n",
    "                i = n - 1\n",
    " \n",
    "        if magicSquare[int(i)][int(j)]:  # 2nd condition\n",
    "            j = j - 2\n",
    "            i = i + 1\n",
    "            continue\n",
    "        else:\n",
    "            magicSquare[int(i)][int(j)] = num\n",
    "            num = num + 1\n",
    " \n",
    "        j = j + 1\n",
    "        i = i - 1  # 1st condition\n",
    " \n",
    "    # Printing magic square\n",
    "    print(\"Magic Square for n =\", n)\n",
    "    print(\"Sum of each row or column\",\n",
    "          n * (n * n + 1) // 2, \"\\n\")\n",
    " \n",
    "    for i in range(0, n):\n",
    "        for j in range(0, n):\n",
    "            print('%2d ' % (magicSquare[i][j]),\n",
    "                  end='')\n",
    " \n",
    "            # To display output\n",
    "            # in matrix form\n",
    "            if j == n - 1:\n",
    "                print()\n",
    " \n",
    "# Driver Code\n",
    " \n",
    " \n",
    "# Works only when n is odd\n",
    "n = 7\n",
    "generateSquare(n)"
   ]
  },
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    "# Honest and Liers"
   ]
  },
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    "Suppose there 16 People sit around a circular table.  ٍEach claims that the two people next to him/her are liars. How many liars are there in the crowd? "
   ]
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