{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "20fce00f",
   "metadata": {},
   "source": [
    "# ML, Data Analysis\n",
    "### Probability: Expected value \n",
    "\n",
    "The **expected value** (also called the **expectation** or **mean**) is a fundamental concept in probability theory and statistics. It represents the long-run average value of a random variable if an experiment is repeated many times.\n",
    "<hr>\n",
    "\n",
    "For a **discrete random variable** $X$, the expected value $E[X]$ is defined as:\n",
    "<br>$E[X]=∑_{x_i} x_i⋅P(X=x_i)$\n",
    "<br>where $P(X=x_i)$ is the probability of the random variable $X$ takes the value $x_i$.\n",
    "<br>Or we can define the expected value with the probability (mass) function $f(x)$ of $X$:\n",
    "<br>$E[X]=∑_x x⋅f(x)$\n",
    "<hr>\n",
    "\n",
    "For a **continuous random variable** $X$, the expected value $E[X]$ is defined as:\n",
    "<br>$E[X]=\\int_{-\\infty}^{\\infty} x\\cdot f(x)dx$\n",
    "<br>where $f(x)$ is the probability (density) function (PDF) of $X$.\n",
    "<hr>\n",
    "\n",
    "**Contents:**\n",
    " - Computing the expected value for a discrete random variable\n",
    " - Computing the expected value for a continuous random variable\n",
    "<hr>\n",
    "https://github.com/ostad-ai/Machine-Learning\n",
    "<br> Explanation: https://www.pinterest.com/HamedShahHosseini/Machine-Learning/background-knowledge"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "f0c89242",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Import thr required modules\n",
    "import numpy as np\n",
    "from matplotlib import pyplot as plt\n",
    "# Import the required function for integration\n",
    "from scipy.integrate import quad"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "ac135579",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Expected value of rolling a fair six-sided die: 3.5\n"
     ]
    }
   ],
   "source": [
    "# Example: discrete random variable\n",
    "# X represents the outcome of rolling a fair six-sided die\n",
    "# Define the outcomes and their probabilities\n",
    "outcomes = [1, 2, 3, 4, 5, 6]\n",
    "# Each outcome has an equal probability (here, 1/6)\n",
    "probabilities = [1/len(outcomes)] * len(outcomes)  \n",
    "\n",
    "# Calculate the expected value\n",
    "expected_value = sum(x * p for x, p in zip(outcomes, probabilities))\n",
    "\n",
    "print(f\"Expected value of rolling a fair six-sided die: {expected_value}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "6e5c163f",
   "metadata": {},
   "outputs": [
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# let's draw the probability (mass) function for example above:\n",
    "plt.bar(outcomes,probabilities,.05,color='orange')\n",
    "plt.scatter(outcomes,probabilities,s=50,c='k')\n",
    "# The expected value is shown with red mark\n",
    "plt.scatter(expected_value,0,s=200,c='red',marker='o')\n",
    "plt.title('Probability (mass) function for the discrete example of a fair die')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7dd1a6a5",
   "metadata": {},
   "source": [
    "<hr style=\"height:5px; background-color:green\">"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "d70cec57",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Expected value of the continuous random variable: 0.6666666666666666\n"
     ]
    }
   ],
   "source": [
    "# Example: continuous random variable\n",
    "# Define the probability (density) function (for short, PDF)\n",
    "def pdf(x):\n",
    "    return 2. * x if 0 <= x <= 1 else 0.\n",
    "\n",
    "# Define the integrand for the expected value: x * f(x)\n",
    "def integrand(x):\n",
    "    return x * pdf(x)\n",
    "\n",
    "# Compute the expected value using numerical integration\n",
    "expected_value, _ = quad(integrand, -np.inf, np.inf)\n",
    "\n",
    "print(f\"Expected value of the continuous random variable: {expected_value}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "74ae0061",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# let's draw the probability (density) function for example above:\n",
    "xs=np.linspace(-1,2,100)\n",
    "pdf_vec=np.vectorize(pdf)\n",
    "ys=pdf_vec(xs)\n",
    "plt.plot(xs,ys,'-')\n",
    "# The expected value is shown with red mark\n",
    "plt.scatter(expected_value,0,s=100,c='red',marker='o')\n",
    "plt.title('Probability (density) function of the continuous example')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "d47e820d",
   "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": "3.11.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}