From 96d5314edf560d93fb96eedaed249d13805e5bb6 Mon Sep 17 00:00:00 2001 From: Caleb Burke Date: Thu, 28 May 2026 01:04:32 -0700 Subject: [PATCH] Chapter 3 progress --- .../README.md | 2 +- .../ch2/example-problems.ipynb | 57 ---------------- .../ch3/example-problems.ipynb | 42 ++++++++++++ .../ch3/notes.ipynb | 67 ++++++++++++++++++- 4 files changed, 107 insertions(+), 61 deletions(-) delete mode 100644 study/001_introduction-to-probability-statistics-and-random-processes/ch2/example-problems.ipynb diff --git a/study/001_introduction-to-probability-statistics-and-random-processes/README.md b/study/001_introduction-to-probability-statistics-and-random-processes/README.md index 819bd0e..f8418b8 100644 --- a/study/001_introduction-to-probability-statistics-and-random-processes/README.md +++ b/study/001_introduction-to-probability-statistics-and-random-processes/README.md @@ -4,7 +4,7 @@ total pages=1007 -**Currently reading:** chapter 3, page 157 +**Currently reading:** chapter 3, page 160 TODO: diff --git a/study/001_introduction-to-probability-statistics-and-random-processes/ch2/example-problems.ipynb b/study/001_introduction-to-probability-statistics-and-random-processes/ch2/example-problems.ipynb deleted file mode 100644 index 3ee666f..0000000 --- a/study/001_introduction-to-probability-statistics-and-random-processes/ch2/example-problems.ipynb +++ /dev/null @@ -1,57 +0,0 @@ -{ - "cells": [ - { - "cell_type": "code", - "execution_count": 1, - "id": "206bf674", - "metadata": {}, - "outputs": [], - "source": [ - "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\")" - ] - }, - { - "cell_type": "markdown", - "id": "612bd02c", - "metadata": {}, - "source": [ - "# Chapter 2 Example Problems" - ] - }, - { - "cell_type": "markdown", - "id": "b7f23a4f", - "metadata": {}, - "source": [] - } - ], - "metadata": { - "kernelspec": { - "display_name": "roadmap (3.14.5)", - "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": "3.14.5" - } - }, - "nbformat": 4, - "nbformat_minor": 5 -} diff --git a/study/001_introduction-to-probability-statistics-and-random-processes/ch3/example-problems.ipynb b/study/001_introduction-to-probability-statistics-and-random-processes/ch3/example-problems.ipynb index 0fa0e69..9514368 100644 --- a/study/001_introduction-to-probability-statistics-and-random-processes/ch3/example-problems.ipynb +++ b/study/001_introduction-to-probability-statistics-and-random-processes/ch3/example-problems.ipynb @@ -25,6 +25,48 @@ "source": [ "# Chapter 3 Example Problems" ] + }, + { + "cell_type": "markdown", + "id": "60ec2620", + "metadata": {}, + "source": [ + "## Example 3.4\n", + "\n", + "$$ R_Y = \\mathbb{N} $$" + ] + }, + { + "cell_type": "code", + "execution_count": 26, + "id": "f1631c9c", + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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", 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" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "p = 0.4\n", + "y = np.arange(1, 11) # geometric starts at 1\n", + "pmf = ((1 - p)**(y - 1)) * p\n", + "\n", + "plt.stem(y, pmf)\n", + "plt.title(r\"$P_Y(y) = (1-p)^{y-1}p$, where $p=$\" + f\"{p}\")\n", + "plt.xlabel(r\"$y$\")\n", + "plt.ylabel(r\"$P_Y(y)$\")\n", + "plt.xticks(y)\n", + "plt.ylim(bottom=0)\n", + "plt.tight_layout()\n", + "plt.show()" + ] } ], "metadata": { diff --git a/study/001_introduction-to-probability-statistics-and-random-processes/ch3/notes.ipynb b/study/001_introduction-to-probability-statistics-and-random-processes/ch3/notes.ipynb index 3f15e9d..58c31e2 100644 --- a/study/001_introduction-to-probability-statistics-and-random-processes/ch3/notes.ipynb +++ b/study/001_introduction-to-probability-statistics-and-random-processes/ch3/notes.ipynb @@ -31,14 +31,75 @@ "id": "9f0046c2", "metadata": {}, "source": [ - "## Random Variables (3.1.1 - 3.1.2)\n", - "\n", + "## Random Variables (3.1.1 - 3.1.2)" + ] + }, + { + "cell_type": "markdown", + "id": "7c73b89f", + "metadata": {}, + "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", - "***Definition.*** $X$ is a discrete random variable, if its range is countable\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" ] + }, + { + "cell_type": "markdown", + "id": "afa12aa6", + "metadata": {}, + "source": [ + "***Definition.*** $X$ is a discrete random variable, if its range is countable" + ] + }, + { + "cell_type": "markdown", + "id": "abd4a3ea", + "metadata": {}, + "source": [ + "### 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" + ] + }, + { + "cell_type": "markdown", + "id": "a7dab8e8", + "metadata": {}, + "source": [ + "***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 $P_X(x) = 0$ " + ] + }, + { + "cell_type": "markdown", + "id": "84ca67da", + "metadata": {}, + "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 $A \\subset R_X, P(X \\in A) = \\sum_{x \\in A} P_X(x)$" + ] + }, + { + "cell_type": "markdown", + "id": "57323d97", + "metadata": {}, + "source": [] } ], "metadata": {