FirstSession.ipynb

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    "# First Session of Stochastic Modelling\n",
    "[![Open in Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/guglielmopadula/StochasticModelling2022-2023/blob/main/FirstSession.ipynb)\n",
    "\n",
    "## Part 1: A very fast recap of Probability\n",
    "\n",
    "From Wikipedia:\n",
    "\n",
    "The **probability** of an event is a number that indicates how likely the event is to occur.\n",
    "\n",
    "### Basic rules of probability:\n",
    "\n",
    "$$P(A\\cup B)+P(A\\cap B)=P(A)+P(B) \\quad \\forall A,B \\in \\Omega$$\n",
    "\n",
    "$$0 \\le P(A)\\le 1 \\quad \\forall A \\in \\Omega$$ \n",
    "\n",
    "$$P(A|B)P(B)=P(A\\cap B) \\quad \\forall A,B \\in \\Omega$$ (Bayes law)\n",
    "\n",
    "$$P(A)=P(A|B)P(B)+P(A|B^{C})P(B^{C}) \\quad \\forall A,B \\in \\Omega$$ (law of total probability)\n",
    "\n",
    "$$P(B^{C})=1-P(B) \\quad \\forall B\\in \\Omega$$\n",
    "\n",
    "If $A$ and $B$ are independent we have $$P(A \\cap B)= P(A)P(B) \\quad \\forall A,B \\in \\Omega$$\n",
    "\n",
    "### A realistic exercise\n",
    "\n",
    "Let's do a very basic exercise (which I actually did in the last year):\n",
    "\n",
    "It is known from the academic year 2020-2021 that a student who does his Stochastic Modelling homeworks has a chance of 90 percent to get a very good grade (between $27$ and $30$); but the chance drops to 75 percent if he doesn't do the homeworks.\n",
    "Guglielmo has been very busy with other courses and figures out that he has only a 69 percent chance of doing the homeworks.\n",
    "What are his chances of not getting a very good grade in the course?\n",
    "\n",
    "Answer:\n",
    "There are two important events under discussion. Let us name them.\n",
    "\n",
    "$E$ : doing homeworks.\n",
    "\n",
    "$F$ : getting a very good grade.\n",
    "\n",
    "We know that $$P(F|E)=0.9$$, $$P(F|E^{C})=0.75$$ and $$P(E)=0.69$$\n",
    "\n",
    "The solution is to use the law of total probability.\n",
    "\n",
    "$$P(F)=P(F|E)P(E)+P(F|E^{C})P(E^{C})$$\n",
    "\n",
    "## Part 2: A very fast recap of Random Variables\n",
    "\n",
    "- A random variable is function $$X: \\Omega \\rightarrow \\mathbb{R}$$ \n",
    "- A random variable is characterized by it's cumulative density function $F(x)=P(X\\le x)$.\n",
    "- A discrete random variable has a countable support and it is characterized by it's probability mass function $P(X=x)=F(x)-F(X=\\max_{y\\in X|F(y)< F(x)} y)$.\n",
    "- A continous random variable has a noncountable support and is characterized by it's density function $f(x)=\\frac{dF}{dx}$\n",
    "\n",
    "The most important properties of a random variable are the moments\n",
    "- non central moments: $E[X^{k}]=\\int_{Im(X)}y^{k}f(y)dy$ if the variable is continuous or $E[X^{k}]=\\sum_{y \\in Im(x)}y^{k}f(y)$ if the variable is discrete\n",
    "- central moments: $E[(X-E[X])^{k}]=\\int_{Im(X)}(y-E[X])^{k}f(y)dy$ if the variable is continuous or $E[X^{k}]=\\sum_{y \\in Im(x)}(y-E[X])^{k}f(y)$ if the variable is discrete\n",
    "- marginalization: $f(x)=\\int_{Im(Y)}f(x,y)dy$ for continuos random variables and $P(X=x)=\\sum_{Im(Y)}P(X=x,Y=y)$\n",
    "- conditioning $f(x|y)=\\frac{f(x,y)}{f(y)}$ for continous random variables and $P(X=x|Y=y)=\\frac{P(X=x,Y=y)}{P(Y=y)}$ for discrete random variables\n",
    "\n",
    "Examples of r.v:\n",
    "- continuous: Gaussian, Exponential, Beta, UniformContinous\n",
    "- discrete: Bernoulli, Binomial, Poisson\n",
    "\n",
    "\n",
    "Properties of mean and variance:\n",
    "$$E[aX+bY]=aE[X]+bE[Y]$$\n",
    "$$Var[aX]=a^{2}Var[X]$$\n",
    "$$Var[X+Y]=Var[X]+Var[Y]+2Cov(X,Y)$$\n",
    "\n",
    "Some random variables are identically distributed if they have the same cdf/pdf/pgh.\n",
    "There are two very important theorems in probability. Let's assume that we have a set of i.i.d  random variables $X_{i}$ with finite mean and variance, then:\n",
    "- Central limit theorem: $$\\frac{\\sum\\limits_{i=1}^{n} X_{i}-nE[X]}{\\sqrt{nVar[X]}}\\rightarrow Normal(0,1)$$\n",
    "- Law of large numbers: $$\\frac{\\sum\\limits_{i=1}^{n} X_{i}}{n}\\rightarrow E[X]$$\n",
    "\n",
    "## A coin tossing (AKA Monte Carlo Integral)\n",
    "Let's suppose we want to see if a coin is balanced. We may use random variables. We may toss a coin a lot of times and assign to a random variable the value 1 is we get head and the value 0 if we get tail. So we get a binary variable, which is usually called Bernoulli random variable. \n",
    "$$X=\\begin{cases} 1 \\text{ with probability } p \\\\\n",
    "0 \\text{ with probability } 1-p \\end{cases}$$\n",
    "\n",
    "An important property of a Bernoulli random variable is that $E[X]=0*(1-p)+1*p=p$\n",
    "\n",
    "Our coin is balanced if and only if we have $p=0.5$.\n",
    "\n",
    "As tossing a coin is slow (and I only have one hour) we are going to simulate the tosses using scipy."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "24f785fd",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tail\n"
     ]
    }
   ],
   "source": [
    "#!pip  numpy matplotlib scipy dill\n",
    "from scipy.stats import bernoulli #basic python importing\n",
    "from tqdm import trange #Very useful function that you probably don't know\n",
    "class TossingMachine():\n",
    "    def __init__(self):\n",
    "        self.p=0.5\n",
    "        self.dist=bernoulli\n",
    "    \n",
    "    def toss(self):\n",
    "        a=self.dist.rvs(self.p)\n",
    "        if a==1:\n",
    "            return \"Head\"\n",
    "        else:\n",
    "            return \"Tail\"\n",
    "\n",
    "tossing_machine=TossingMachine()\n",
    "print(tossing_machine.toss())"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c8adad7e",
   "metadata": {},
   "source": [
    "To see if our virtual coin is balanced or not we may toss a lot of times and then calculate the sample avarage. \n",
    "The sample average will approximate our probability $p$ because of the law of large numbers.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "8a9e1b8c",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "100%|████████████████████████████████████████████████████████████████████████████████████████████████████████████| 100000/100000 [00:03<00:00, 25901.15it/s]"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "the estimated probability is  0.49794\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fb3550aa5e0>]"
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     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "X=[] #inizialization of the list\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "NUM_SAMPLES=100000\n",
    "for i in trange(NUM_SAMPLES):\n",
    "    tmp=tossing_machine.toss()\n",
    "    if tmp==\"Head\":\n",
    "        X.append(1)\n",
    "    else:\n",
    "        X.append(0)\n",
    "\n",
    "print(\"the estimated probability is \", sum(X)/NUM_SAMPLES)\n",
    "avarage_mean=np.cumsum(X)/np.cumsum(np.ones(NUM_SAMPLES))\n",
    "plt.plot(np.arange(NUM_SAMPLES)+1,avarage_mean)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dda3db64",
   "metadata": {},
   "source": [
    "We got a value near $0.5$. So we are happy :-) ."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "93b6224f",
   "metadata": {},
   "source": [
    "## The queen of all r.v: the Gaussian R.V. \n",
    "### but republic is approaching ....\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "08afd764",
   "metadata": {},
   "source": [
    "The Gaussian distribution is characterized by it's mean $\\mu$ and it's variance $\\sigma^{2}$.\n",
    "It has the following pdf\n",
    "$$\n",
    "f(x)=\\frac{1}{\\sqrt{2\\pi}\\sigma}e^{-(\\frac{x-\\mu}{\\sigma})^{2}}$$\n",
    "The parameter $\\sigma$ is called standard deviation. \n",
    "The Gaussian distribution is particulary important because of the following two properties:\n",
    "- it is the limit distribution of the central limit theorem\n",
    "- it is the only distribution with finite variance such that all linear combinations of **independent** random variables from it's distribution have it's distribution:\n",
    "$$ X_{i}\\sim N(\\mu_{i},\\sigma_{i}^{2})$$ \n",
    "$$\\sum_{i=1}^{n} a_{i}X_{i}\\sim N(\\sum\\limits_{i=1}^{n} a_{i}\\mu_{i},\\sum\\limits_{i=1}^{n} a_{i}^{2}\\sigma_{i}^{2})$$\n",
    "These two properties are strongly linked with each other, for a proof of this I invite to follow the course of Information Theory.\n",
    "\n",
    "Let's now look at scipy stats function.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "019bf0bb",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "P(X<=0) is,  0.4012936743170763\n",
      "Variance of X is 16.0\n",
      "Mean of X is 1.0\n",
      "f(0) is 0.09666702920071232\n",
      "Sampling from Normal is -1.5705329900003977\n"
     ]
    }
   ],
   "source": [
    "from scipy.stats import norm #normal random variable\n",
    "distribution=norm(1,4) #mean=1, standard deviation=4,not variance!!!\n",
    "print(\"P(X<=0) is, \",distribution.cdf(0))\n",
    "print(\"Variance of X is\", distribution.var())\n",
    "print(\"Mean of X is\", distribution.mean())\n",
    "print(\"f(0) is\",distribution.pdf(0))\n",
    "print(\"Sampling from Normal is\", distribution.rvs())"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "eb70331d",
   "metadata": {},
   "source": [
    "What is the standard deviation of the sum of two independent normal random variable of standard deviation 3 and 4? 5 because you have to do $\\sqrt{3^{2}+4^{2}}=5$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e605a076",
   "metadata": {},
   "source": [
    "## Notable discrete random variables\n",
    "$$\n",
    "\\begin{array}{|c|c|c|c|c|c|}\n",
    "\\hline x & x \\text { Counts } & P.M.F & \\text { Values of } X & E(x) & V(x) \\\\\n",
    "\\hline \\begin{array}{l}\n",
    "\\text { Discrete } \\\\\n",
    "\\text { uniform }\n",
    "\\end{array} & \\begin{array}{l}\n",
    "\\text { Outcomes that are } \\\\\n",
    "\\text { equally likely (finite) }\n",
    "\\end{array} & \\frac{1}{b-a+1} & a \\leq x \\leq b & \\frac{b+a}{2} & \\frac{(b-a+2)(b-a)}{12} \\\\\n",
    "\\hline \\text { Bernoulli } & \\begin{array}{l}\n",
    "\\text { Number of sucesses } \\\\\n",
    "\\text { in }  \\text { 1 fixed trial }\n",
    "\\end{array} & \\left(\\begin{array}{l}\n",
    "n \\\\\n",
    "x\n",
    "\\end{array}\\right) p^{x}(1-p)^{1-x} & x=0,1 & p & p(1-p) \\\\\n",
    "\\hline \\text { Binomial } & \\begin{array}{l}\n",
    "\\text { Number of sucesses } \\\\\n",
    "\\text { in } n \\text { fixed trials }\n",
    "\\end{array} & \\left(\\begin{array}{l}\n",
    "n \\\\\n",
    "x\n",
    "\\end{array}\\right) p^x(1-p)^{n-x} & x=0,1, \\ldots, n & np & n p(1-p) \\\\\n",
    "\\hline \\text { Poisson } & \\begin{array}{l}\n",
    "\\text { Number of arrivals in } \\\\\n",
    "\\text { a fixed time period }\n",
    "\\end{array} & \\frac{e^{-\\lambda} \\lambda^x}{x !} & x=0,1,2, \\ldots & \\lambda & \\lambda \\\\\n",
    "\\hline \\text { Geometric } & \\begin{array}{l}\n",
    "\\text { Number of trials up } \\\\\n",
    "\\text { through 1st success }\n",
    "\\end{array} & (1-p)^{x-1} p & x=1,2,3, \\ldots & \\frac{1}{p} & \\frac{1-p}{p^2} \\\\\n",
    "\\hline \\begin{array}{l}\n",
    "\\text { Negative } \\\\\n",
    "\\text { Binomial }\n",
    "\\end{array} & \\begin{array}{l}\n",
    "\\text { Number of trials up } \\\\\n",
    "\\text { through kth success }\n",
    "\\end{array} & \\left(\\begin{array}{l}\n",
    "x-1 \\\\\n",
    "k-1\n",
    "\\end{array}\\right)(1-p)^{x-k} p^k & 0^k \\quad x=k, k+1, \\ldots & \\frac{k}{p} & \\frac{k(1-p)}{p^2} \\\\\n",
    "\\hline \\begin{array}{c}\n",
    "\\text { Hyper - } \\\\\n",
    "\\text { geometric }\n",
    "\\end{array} & \\begin{array}{l}\n",
    "\\text { Number of marked } \\\\\n",
    "\\text { individuals in } \\\\\n",
    "\\text { sample taken } \\\\\n",
    "\\text { without replacement }\n",
    "\\end{array} & \\frac{\\left(\\begin{array}{l}\n",
    "M \\\\\n",
    "x\n",
    "\\end{array}\\right)\\left(\\begin{array}{l}\n",
    "N-M \\\\\n",
    "n-x\n",
    "\\end{array}\\right)}{(N)} & \\begin{array}{c}\n",
    "\\max (0, M+n-N) \\\\\n",
    "\\leq x \\leq \\min (M, n)\n",
    "\\end{array} & n * \\frac{M}{N} & \\frac{\\mathrm{nM}(N-M)(N-n)}{N^2(N-1)} \\\\\n",
    "\\hline\n",
    "\\end{array}\n",
    "$$\n",
    "\n",
    "## Notable Continous distribution\n",
    "$$\n",
    "\\begin{array}{cccccc}\n",
    "\\hline X & X \\text { Measures } & P.D.F & \\text { Values of } X & E(x) & \\boldsymbol{V}(\\mathbf{x}) \\\\\n",
    "\\hline \\begin{array}{c}\n",
    "\\text { Continuous } \\\\\n",
    "\\text { uniform }\n",
    "\\end{array} & \\begin{array}{c}\n",
    "\\text { Outcomes with equal } \\\\\n",
    "\\text { density (continuous) }\n",
    "\\end{array} & \\frac{1}{\\mathrm{~b}-\\mathrm{a}} & \\mathrm{a} \\leq \\mathrm{x} \\leq \\mathrm{b} & \\frac{\\mathrm{b}+\\mathrm{a}}{2} & \\frac{(\\mathrm{b}-\\mathrm{a})^2}{12} \\\\\n",
    "\\hline \\text { Exponential } & \\begin{array}{c}\n",
    "\\text { Time between events; } \\\\\n",
    "\\text { time until an event }\n",
    "\\end{array} & \\lambda \\mathrm{e}^{-\\lambda x} & x \\geq 0 & \\frac{1}{\\lambda} & \\frac{1}{\\lambda^2} \\\\\n",
    "\\hline \\text { Erlang } & \n",
    "\\text { time until k events occur}\n",
    " & \\frac{\\lambda^k x^{k-1} e^{-\\lambda x}}{(k-1) !} & x \\geq 0 & \\frac{k}{\\lambda} & \\frac{k}{\\lambda^2} \\\\\n",
    "\\hline \\text { Normal } & \\begin{array}{c}\n",
    "\\text { Values with a bell-shaped } \\\\\n",
    "\\text { distribution (continuous) }\n",
    "\\end{array} & \\frac{1}{\\sigma \\sqrt{2 \\pi}} \\mathrm{e}^{-\\frac{1}{2}\\left(\\frac{x-\\mu}{\\sigma}\\right)^2} & -\\infty<\\mathrm{x}<\\infty & \\mu & \\sigma \\\\\n",
    "\\hline \\text { Beta } & \n",
    "\\text {order statistics}\n",
    " & \\frac{\\Gamma(\\alpha+\\beta)x^{\\alpha-1}(1-x)^{\\beta-1}}{\\Gamma(\\alpha)\\Gamma(\\beta)}& 0 \\le x \\le 1 &\\frac{\\alpha}{\\alpha+\\beta} & \\frac{\\alpha \\beta}{(\\alpha+\\beta)^2(\\alpha+\\beta+1)}\\\\\n",
    "\\hline \\text { Gamma } & \n",
    "\\text {I have no idea}\n",
    " & \\frac{1}{\\Gamma(k) \\theta^k} x^{k-1} e^{-\\frac{x}{\\theta}}& x\\ge 0 &k\\theta & k\\theta^{2}\\\\\n",
    "\\hline\n",
    "\\end{array}\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f5a7c229",
   "metadata": {},
   "source": [
    "## Histogram and Kernel Density Estimation"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ce678870",
   "metadata": {},
   "source": [
    "Suppose that you have a blackbox random variable from which you can sample and you want to estimate it's pdf.\n",
    "\n",
    "You could start by plotting the histogram. To plot an histogram we need to choose a number of bins:\n",
    "- with a small number of bins there is a very low resolution\n",
    "- with a high number of bins there is a lot of variance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "10f85556",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import dill\n",
    "import scipy\n",
    "import matplotlib.pyplot as plt\n",
    "blackboxrv=dill.load(open( \"blackbox.p\", \"rb\" ))\n",
    "dataset=blackboxrv.sample(50000)\n",
    "a=plt.hist(dataset,density=True,bins=2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "e9b9f2da",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "dataset=blackboxrv.sample(50000)\n",
    "a=plt.hist(dataset,density=True,bins=10000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "id": "eee38f68",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "dataset=blackboxrv.sample(50000)\n",
    "a=plt.hist(dataset,density=True,bins=\"scott\") #use scott optimal binning"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "256bacc4",
   "metadata": {},
   "source": [
    "The random variable is clearly bimodal. Let's try to estimate the pdf using a KDE estimation, which basically interpolates using a lot of gaussian distributions using kernels (for details in this technique see the Unsupervised Learning course)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "id": "59957ab6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "epsilon=0.01\n",
    "kde=scipy.stats.gaussian_kde(dataset)\n",
    "a=plt.hist(dataset,density=True,bins=\"scott\")\n",
    "x=np.min(dataset)-epsilon+(np.max(dataset)-np.min(dataset)+2*epsilon)*np.arange(1000)/1000\n",
    "b=plt.plot(x,kde.pdf(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9681b588",
   "metadata": {},
   "source": [
    "The KDE estimate is good.\n",
    "\n",
    "Why is the probability of -0.5 greater then one? Because this is the pdf and it's not a probability."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "21802f4a",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "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": "3.9.16"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}