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    "# ML, Data Analysis\n",
    "### Probability: variance \n",
    "\n",
    "The **variance** measures the spread of the given random variable, which is defined by the following formula:\n",
    "<br>$\\large var(X)=E[(X-E[X])^2]$\n",
    "<br> where $E[X]$ denotes the *expected value* of random variable $X$. We have talked before about the **expected value**.\n",
    "<br>**Hint:** We also call $E[X]$ the **mean** of $X$.\n",
    "<br>\n",
    "<br>Some properties of variance:\n",
    "1. **Non-negativity:** For any random varibale $X$: $var(X)>=0$\n",
    "2. **Linear transformation:** For constants $a$ and $b$, we have: $var(aX+b)=a^2var(X)$\n",
    "3. **Sum of variances:** Having *uncorrelated* random variables $X$ and $Y$ ($E[XY]=E[X]E[Y]$): we have: $var(X+Y)=var(X)+var(Y)$\n",
    "<hr>\n",
    "\n",
    "**Contents:**\n",
    " - Computing the mean and variance for a discrete random variable.\n",
    " - Computing the mean and variance 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"
   ]
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  {
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   "execution_count": 1,
   "id": "f0c89242",
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   "source": [
    "# Import the required function for integration\n",
    "from scipy.integrate import quad"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "d4430b1d",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Functions for discerete random variables\n",
    "def mean(data,probs=None):\n",
    "    n=len(data)\n",
    "    if probs is None:\n",
    "        probs=[1/n]*n\n",
    "    return sum(x*p for x,p in zip(data,probs))\n",
    "\n",
    "def variance(data,probs=None):\n",
    "    mu = mean(data,probs)\n",
    "    n=len(data)\n",
    "    if probs is None:\n",
    "        probs=[1/n]*n\n",
    "    squared_diff = [p*(x - mu) ** 2 for x,p in zip(data,probs)]\n",
    "    return sum(squared_diff)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "eb2a6b01",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The outcomes: [1, 2, 3, 4, 5, 6]\n",
      "The mean: 3.5\n",
      "The variance: 2.9166666666666665\n"
     ]
    }
   ],
   "source": [
    "# Example: discrete random variable\n",
    "outcomes = [1, 2, 3, 4, 5, 6]\n",
    "print(f'The outcomes: {outcomes}')\n",
    "print(f'The mean: {mean(outcomes)}')\n",
    "print(f'The variance: {variance(outcomes)}')"
   ]
  },
  {
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   "id": "1c730379",
   "metadata": {},
   "source": [
    "<hr style=\"height:5px;background-color:lightgreen\">"
   ]
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  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "e97de675",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Functions for 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",
    "def mean_c():\n",
    "    integrand=lambda x: x*pdf(x)\n",
    "    mean, _ = quad(integrand, 0,1)\n",
    "    return mean\n",
    "\n",
    "def variance_c():\n",
    "    mu=mean_c()\n",
    "    integrand=lambda x: (x-mu)**2*pdf(x)\n",
    "    var,_=quad(integrand, 0,1)\n",
    "    return var"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "d47e820d",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The pdf is a ramp defined between zero and one\n",
      "The mean: 0.6666666666666667\n",
      "The variance: 0.05555555555555555\n"
     ]
    }
   ],
   "source": [
    "# Example: continuous random variable\n",
    "print('The pdf is a ramp defined between zero and one')\n",
    "print(f'The mean: {mean_c()}')\n",
    "print(f'The variance: {variance_c()}')"
   ]
  },
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   "cell_type": "code",
   "execution_count": null,
   "id": "e4203044",
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   "outputs": [],
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