study/study/001_introduction-to-probability-statistics-and-random-processes/ch3/notes.ipynb

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    "import os\n",
    "import sys\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import seaborn as sns\n",
    "\n",
    "sns.set_theme(style=\"whitegrid\", context=\"notebook\")"
   ]
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   "cell_type": "markdown",
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   "source": [
    "# Chapter 3 Notes"
   ]
  },
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   "source": [
    "## Random Variables (3.1.1 - 3.1.3)"
   ]
  },
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   "source": [
    "***Definition.*** Random Variables: \\\n",
    "A random variable $X$ is a function from the sample space to the real numbers. ie\n",
    "$$X : S \\to \\mathbb{R}$$\n",
    "\n",
    "Each outcome ($\\omega$) in the sample space must have a $X(\\omega)$ defined. \n",
    "\n",
    "> Note that $X$ is a deterministic function. The randomness comes from the fact that we dont know the inputs to $X$, ie the outcome of the random experiment"
   ]
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  {
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   "source": [
    "***Definition.*** $X$ is a discrete random variable, if its range is countable"
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    "### More on $X$\n",
    "\n",
    "Let $\\Omega$ be a sample space and let $X$ be a random variable on $\\Omega$\n",
    "\n",
    "- $X$ is a function, and $\\forall \\omega \\in \\Omega, X(\\omega) \\in \\mathbb{R}\\quad$ (ie $X(\\omega)$ is defined on all **outcomes**) \n",
    "- $P_x(1)$ is asking: For event $A = \\{\\omega \\in \\Omega \\mid X(\\omega) = 1 \\}$, what is $P(A)$?\n",
    "- $X$ induces a partition of $\\Omega$\n"
   ]
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    "***Definition.*** Let $X$ be a discrete random variable with range $R_X = \\{x_1, x_2, x_3, \\dots\\}$ (finite or countably infinite). The function\n",
    "\n",
    "$$P_X(x_k) = P(X = x_k), \\text{for} k = 1,2,3,\\dots,$$\n",
    "\n",
    "is called the *probability mass function (PMF)* of $X$. (also called the probability distribution)\n",
    "\n",
    "Note that if $x \\notin R_X$, then $P_X(x) = 0$ "
   ]
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   "source": [
    "## Properties of PMF\n",
    "- $0 \\geq P_X(x) \\geq 1, \\forall x$\n",
    "- $\\sum_{x \\in R_X}P_X(x) = 1$ \n",
    "- for any set $A \\subset R_X, P(X \\in A) = \\sum_{x \\in A} P_X(x)$"
   ]
  },
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   "metadata": {},
   "source": [
    "## Independent Random Variables (3.1.4)"
   ]
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   "metadata": {},
   "source": [
    "***Definition.*** Consider two discrete random variables $X$ and $Y$. We say that $X$ and $Y$ are independent if:\n",
    "\n",
    "$$P(X=x, Y=y) = P(X=x)P(Y=y)$$\n",
    "\n",
    "In general, if two random variables are independent, then you can write\n",
    "\n",
    "$$P(X \\in A, Y \\in B) = P(X \\in A)(Y \\in B)$$"
   ]
  },
  {
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   "id": "7b0362c1",
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   "source": [
    "***Definition.*** Consider $n$ discrete random variables $X_1, X_2, X_3, \\dots, X_n$. We say that $X_1, X_2, X_3, \\dots, X_n$ are independent if:\n",
    "\n",
    "$$P(X_1 = x_1, X_2 = x_2, X_3 = x_3, \\dots, X_n = x_n) = P(X_1 = x_1)P(X_2 = x_2) \\dots P(X_n = x_n)$$\n"
   ]
  },
  {
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   "metadata": {},
   "source": [
    "## Special Distributions (3.1.5)"
   ]
  },
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   "metadata": {},
   "source": [
    "***Definition.*** A random variable $X$ is said to be a *Bernoulli* random variable with parameter *p*, shown as $X \\sim \\text{Bernoulli}(p),$ if its PMF is given by"
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