"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"import scipy as sp\n",
"import scipy.stats\n",
"from mpl_toolkits.mplot3d import Axes3D\n",
"import sympy as sy\n",
"sy.init_printing() \n",
"import matplotlib as mpl\n",
"mpl.rcParams['text.latex.preamble'] = r'\\usepackage{amsmath}'"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [],
"source": [
"import warnings\n",
"warnings.filterwarnings(\"ignore\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Perhaps this one of the most important application of linear algebra. We will build up intuition gradually before turning to multivariate normal distribution."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Univariate Normal Distribution"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The PDF of univariate normal distribution is given by\n",
"\n",
"$$\n",
"p(x; \\mu, \\sigma^2) = \\frac{1}{\\sqrt{2\\pi}\\sigma}\\exp{\\left(-\\frac{1}{2\\sigma^2}(x-\\mu)^2\\right)}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"where $p(x; \\mu, \\sigma^2)$ mean random variable is $x$, parameters are $\\mu$ and $\\sigma^2$, it is not a conditional sign which commonly looks like $p(x|y)$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Note that $-\\frac{1}{2\\sigma^2}(x-\\mu)^2$ is a quadratic function, which is the univariate version of quadratic form we have seen in earlier chapters."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"If we define $\\sigma = 2$, $\\mu = 1$, let's plot the quadratic function and its exponential."
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
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