1 Probabilities fundamentals.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Probabilities fundamentals:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib\n",
    "import matplotlib.pyplot as plt\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "%matplotlib inline\n",
    "\n",
    "import seaborn as sns\n",
    "sns.set_style(\"white\")\n",
    "\n",
    "import time\n",
    "\n",
    "import scipy.stats "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem:\n",
    "\n",
    "The joint density function for the random variables ($X,Y$) is given by : \n",
    "\n",
    "$f(x,y)= 10xy^{2}$ ,   $0< x< y< 1 $\n",
    "\n",
    "$= 0  $ elsewhere \n",
    "\n",
    "(a) Find the marginal distribution of $X$ and $Y$ \n",
    "\n",
    "(b) Compute the probability of $0< x< 1/2, 1/4<y< 1/2 $\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Please answer this problem by adding cells here.**$$ $$\n",
    "(a) for marginal distrubution of $X$: $ $ $0<y<1$\n",
    "        $$ f_{Y}(y)= \\int_{-\\infty}^{+\\infty} f(x,y) \\text{d}x = \\int_{0}^{y} 10xy^2 \\text{d}x = 5y^4 $$\n",
    "   for marginal distrubution of $Y$: $ $ $0<x<1$\n",
    "        $$ f_{X}(x)= \\int_{-\\infty}^{+\\infty} f(x,y) \\text{d}y = \\int_{x}^{1} 10xy^2 \\text{d}y = \\frac{10}{3}x(1-x^3) $$\n",
    "        $$ $$\n",
    "        $$ $$\n",
    "(b) the corresponding probability can be expressed as:\n",
    "$$ P= \\iint 10xy^2 \\text{d}x\\text{d}y $$ $$ $$\n",
    "where the integration region is formed by: $ 0<x<\\frac{1}{2}$, $\\frac{1}{4}<y<\\frac{1}{2}$, and $x<y$. Thus, the probability under that condition can be calculated as:<br>$$ $$\n",
    "$$ P(0<x<\\frac{1}{2},\\, \\frac{1}{4}<y<\\frac{1}{2}) = \\int_{\\frac{1}{4}}^{\\frac{1}{2}}\\int_{0}^{\\frac{1}{2}}10xy^2 dy dx =0.045 $$\n",
    "\n",
    "\n",
    "   "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem:\n",
    "\n",
    "Sulfar Dioxide concentrations in the air samples taken in a certain region has been found to be well represented by lognormal distribution with parameters $\\mu = 2.1$ and $\\sigma= 0.8$.\n",
    "\n",
    "(a) What proportions of air samples have concentration between 3 and 6? \n",
    "\n",
    "(b) What proportion do not exceed 10?\n",
    "\n",
    "(c) What interval (a,b) is such that 95% of all air samples have concentration values in this interval, with 2.5% have values less than a, and 2.5% have values exceeding b? \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solve:** $$ $$\n",
    "For these quastions, each one of them will be simulated in Python with numpy, matplotlib, and scipy library, coordinates with probability of Lognormal Distribution. $$ $$\n",
    "Probability probability function of Lognormal:\n",
    "$$ f(x)=\\frac{1}{{\\sqrt{(2\\pi)}}{\\sigma}x}exp(-\\frac{(lnx-{\\mu})^2}{2{\\sigma}^2})  $$\n",
    "Cumulated probability function of Lognormal:\n",
    "$$F(x)=\\phi \\frac{(lnx-{\\mu})}{\\sigma}$$\n",
    "and $\\phi$ is thge cumulative distribution function of the standard normal distribution.$$ $$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x8fa45c0>]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.stats import lognorm\n",
    "from scipy.stats import norm\n",
    "\n",
    "\n",
    "std=0.8\n",
    "mean=2.1\n",
    "x=np.linspace(0,20,10000)\n",
    "dist=lognorm([std],loc=mean)\n",
    "plt.plot(x,dist.pdf(x))\n",
    "plt.plot(x,dist.cdf(x))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(a) $ P_{pdf}(3<X<6)=F_{cdf}(6)-F_{cdf}(3)$. demonstrating in Python: $$ $$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.2446740482064489\n"
     ]
    }
   ],
   "source": [
    "p3=norm.cdf(np.log(3), loc=mean, scale=std)\n",
    "p6=norm.cdf(np.log(6), loc=mean, scale=std) \n",
    "p=p6-p3\n",
    "print(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus, 24.46% of air samples have concentration between 3 and 6. $$ $$\n",
    "$$ $$\n",
    "(b) $ P_{pdf}(X<10)=F_{cdf}(10)$ demonstrating in Python: $$ $$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.5999552852599946\n"
     ]
    }
   ],
   "source": [
    "p10=norm.cdf(np.log(10), loc=mean, scale=std)\n",
    "print(p10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus, 59.99% of samples are not exceed 10. $$ $$\n",
    "$$ $$\n",
    "(c) Use iteration method to find the interval:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(1.7023826223346874, 39.17235183561141)\n"
     ]
    }
   ],
   "source": [
    "la,lb=norm.interval(0.95, loc=mean, scale=std) \n",
    "a=np.exp(la) \n",
    "b=np.exp(lb) \n",
    "print((a,b))\n"
   ]
  }
 ],
 "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": "3.6.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}