probs_n_stats_-_probability distributions.ipynb

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  {
   "source": [
    "# Probability distributions"
   ],
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   "source": [
    "from scipy.stats import lognorm, expon, wei\n",
    "import numpy as np \n",
    "import matplotlib.pyplot as plt \n",
    "%matplotlib inline"
   ]
  },
  {
   "source": [
    "## Continuous distributions"
   ],
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  },
  {
   "source": [
    "### Exponential distribution\n",
    "\n",
    "$\n",
    "f(x) = \\lambda e^{-\\lambda x} \\\\\n",
    "F(x) =  1 - e^{-\\lambda x} \\\\\n",
    "X\\sim \\text{Exp}(1) \\implies \\frac{1}{\\lambda} X ~ \\text{Exp}(\\lambda)\\\\  \n",
    "\\mathbb{E} X = \\int_{0}^\\infty x \\lambda e^{-\\lambda x} dx = \\frac{1}{\\lambda} \\\\  \n",
    "\\mathbb{V}(X) = \\mathbb{E}(X^2)-(\\mathbb{E}X)^2 = \\frac{2}{\\lambda^2}-\\frac{1}{\\lambda^2} = \\frac{1}{\\lambda^2}\n",
    "$\n"
   ],
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\n"
     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "scale = 2\n",
    "x = np.linspace(expon.ppf(0.001, scale=scale), expon.ppf(0.999, scale=scale))\n",
    "plt.plot(x, expon.pdf(x, scale=scale))\n",
    "plt.plot(x, expon.cdf(x, scale=scale))\n",
    "plt.hlines(1, x[0], x[-1], linestyles='dashed')"
   ]
  },
  {
   "source": [
    "### Weibull distribution\n",
    "$\n",
    "f(x) = \\alpha \\lambda x^{\\alpha - 1} \\exp(-\\lambda x^\\alpha)\n",
    "$"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "source": [
    "### Log-normal distribution\n",
    "\n",
    "$\n",
    "\\log X \\sim N(\\mu, \\sigma^2) \\\\\n",
    "f(x) = \\frac{1}{\\sigma x \\sqrt{2\\pi}}\\cdot \\exp\\left(-\\frac{(\\log x - \\mu)^2}{2\\sigma^2}\\right)\n",
    "$"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x28fc7a2e108>]"
      ]
     },
     "metadata": {},
     "execution_count": 26
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "<Figure size 432x288 with 1 Axes>",
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\n"
     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "x = np.linspace(0, 5)\n",
    "plt.plot(x, lognorm.pdf(x, 1, 0, 1))\n",
    "plt.plot(x, lognorm.pdf(x, 0, 1, 2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ]
}