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"cell_type": "markdown",
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"source": [
"# 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"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "f0c89242",
"metadata": {},
"outputs": [],
"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)}')"
]
},
{
"cell_type": "markdown",
"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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