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   "source": [
    "# Distributions and Random Numbers\n",
    "\n",
    "\n",
    "Feng Li\n",
    "\n",
    "School of Statistics and Mathematics\n",
    "\n",
    "Central University of Finance and Economics\n",
    "\n",
    "[feng.li@cufe.edu.cn](mailto:feng.li@cufe.edu.cn)\n",
    "\n",
    "[https://feng.li/statcomp](https://feng.li/statcomp)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ed515448",
   "metadata": {
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   "source": [
    "## Basic concepts of random numbers\n",
    "\n",
    "### Preliminary\n",
    "\n",
    "-   **Pseudo random numbers**\n",
    "\n",
    "    -   an algorithm for generating a sequence of numbers that\n",
    "        approximates the properties of random numbers.\n",
    "\n",
    "    -   The sequence is not truly random in that it is completely\n",
    "        determined by a relatively small set of initial values, called\n",
    "        the PRNG’s state.\n",
    "\n",
    "    -   Pseudo random numbers are important in practice for their\n",
    "        **speed** in number generation and their **reproducibility**."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b6a6c8d8",
   "metadata": {
    "slideshow": {
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   "source": [
    "-   **Random seed**\n",
    "\n",
    "    A random seed (or seed state, or just seed) is a number (or vector)\n",
    "    used to initialize a pseudo random number generator.\n",
    "\n",
    "-   The most important random numbers are from uniform distributed\n",
    "    numbers.\n",
    "\n",
    "    `> runif(n,a,b)`\n",
    "\n",
    "-   Numbers selected from a non-uniform probability distribution can be\n",
    "    generated using a uniform distribution PRNG and a function that\n",
    "    relates the two distributions.\n",
    "\n",
    "-   Assume you have uniformly distributed random numbers from \\[0, 1\\],\n",
    "    how do you extend it to \\[a, b\\]?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "88049805",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
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   "source": [
    "## Continuous random variables\n",
    "\n",
    "### Normal Distribution\n",
    "\n",
    "-   **The normal density function**\n",
    "    $$f(x, \\mu, \\sigma) = \\frac{1}{\\sigma\\sqrt{2\\pi}} e^{-\\frac{(x-\\mu)^2}{2\\sigma^2} }$$\n",
    "\n",
    "    `> dnorm(x,mu,sigma)`\n",
    "\n",
    "    `> dnorm(x,mu,sigma, log=TRUE)`\n",
    "\n",
    "    -   In theory, `dnorm(x,mu,sigma, log=TRUE)==log(dnorm(x,mu,sigma))`\n",
    "        \n",
    "    -   but `dnorm(x,mu,sigma, log=TRUE)` but is more stable for very\n",
    "        large values. Why?\n",
    "\n",
    "    -   **We love logs**."
   ]
  },
  {
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   "execution_count": 6,
   "id": "34388c58",
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       "[1] 0"
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       "-Inf"
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       "-Inf"
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       "-Inf"
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       "[1] -Inf"
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       "-5000.9189385332"
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       "-5000.9189385332"
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       "-5000.9189385332"
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       "[1] -5000.919"
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   "source": [
    "dnorm(100, mean=0, sd=1)\n",
    "log(dnorm(100, mean=0, sd=1))\n",
    "\n",
    "dnorm(100, mean=0, sd=1, log=TRUE)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3dbf69b5",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
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   "source": [
    "-   **The CDF** (Cumulative Distribution Function)\n",
    "    $$\\Phi(x)\\;=\\int_{-\\infty}^x f(t, \\mu, \\sigma) d t$$\n",
    "\n",
    "    `> pnorm(q,mu,sigma)`\n",
    "\n",
    "-   **The quantile** (Given CDF, what is x?), i.e. $\\Phi^{-1}(p)$\n",
    "\n",
    "    `> qnorm(p,mu,sigma)`\n",
    "\n",
    "-   **Random numbers from normal distribution**\n",
    "\n",
    "    `> rnorm(n,mu,sigma)`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "ea776f8c",
   "metadata": {
    "slideshow": {
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   },
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    {
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      "text/plain": [
       "Plot with title “Histogram of z”"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 180,
       "width": 240
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "z = rnorm (100,mean=0, sd=1)\n",
    "hist(z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "de109e18",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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OCHH35QDS5cuNDDw2P27NllZWVCUpE5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\nZYpTAAAAAElFTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 180,
       "width": 240
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x  = seq(-5, 5, 0.1)\n",
    "y = dnorm(x = x, mean = 0, sd = 1)\n",
    "plot(x, y, col = \"blue\", type = \"l\", lwd = 4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "262ed398",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "plot without title"
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   ],
   "source": [
    "y1 = pnorm(q = x, mean = 0, sd = 1)\n",
    "plot(x, y1, col = \"blue\", type = \"l\", lwd = 4)"
   ]
  },
  {
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    "### Other types of continuous distribution\n",
    "\n",
    "-   Distribution Function in R\n",
    "\n",
    "    Student t: `{p,d,q}t`\n",
    "    \n",
    "    Chi squared: `{p,d,q}chi`\n",
    "    \n",
    "    Gamma: `{p,d,q}gamma`\n",
    "    \n",
    "    Exponential `{p,d,q}exp`\n",
    "\n",
    "\n",
    "-   For a significance test, what distribution do you use?\n"
   ]
  },
  {
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   "id": "f99866f8",
   "metadata": {
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   "source": [
    "# Discrete random variables\n",
    "\n",
    "### Discrete random variables\n",
    "\n",
    "-   Distribution Function in R\n",
    "\n",
    "    Binomial: `{p,d,q}binorm`\n",
    "    \n",
    "    Negative binomial: `{p,d,q}nbinom`\n",
    "\n",
    "    Poisson: `{p,d,q}pois`\n",
    "    \n",
    "    Geometric: `{p,d,q}geom`\n",
    "    \n",
    "-   Bernoulli distribution is a special case of binomial distribution."
   ]
  },
  {
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   "id": "2c3bbedf",
   "metadata": {
    "slideshow": {
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   "source": [
    "### Suggested reading\n",
    "\n",
    "-   Jones (2009): **Chapter 14, 15, 16**"
   ]
  }
 ],
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