diff --git a/study/001_introduction-to-probability-statistics-and-random-processes/ch4/example-problems.ipynb b/study/001_introduction-to-probability-statistics-and-random-processes/ch4/example-problems.ipynb deleted file mode 100644 index 1d74d04..0000000 --- a/study/001_introduction-to-probability-statistics-and-random-processes/ch4/example-problems.ipynb +++ /dev/null @@ -1,51 +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 4 Example Problems" - ] - } - ], - "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/ch4/notes.ipynb b/study/001_introduction-to-probability-statistics-and-random-processes/ch4/notes.ipynb deleted file mode 100644 index b3cd03a..0000000 --- a/study/001_introduction-to-probability-statistics-and-random-processes/ch4/notes.ipynb +++ /dev/null @@ -1,59 +0,0 @@ -{ - "cells": [ - { - "cell_type": "code", - "execution_count": 2, - "id": "c58309b2", - "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": "a6732353-51d5-4478-9cf8-5834e57e5a4e", - "metadata": {}, - "source": [ - "# Chapter 4 Notes" - ] - }, - { - "cell_type": "markdown", - "id": "9f0046c2", - "metadata": {}, - "source": [ - "## Continuous Random Variables and their Distributions (4.0.0 - 4.0.0)" - ] - } - ], - "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/.ipynb_checkpoints/ch1-checkpoint.ipynb b/textbooks/001_introduction-to-probability-statistics-and-random-processes/.ipynb_checkpoints/ch1-checkpoint.ipynb similarity index 100% rename from study/001_introduction-to-probability-statistics-and-random-processes/.ipynb_checkpoints/ch1-checkpoint.ipynb rename to textbooks/001_introduction-to-probability-statistics-and-random-processes/.ipynb_checkpoints/ch1-checkpoint.ipynb diff --git a/study/001_introduction-to-probability-statistics-and-random-processes/README.md b/textbooks/001_introduction-to-probability-statistics-and-random-processes/README.md similarity index 85% rename from study/001_introduction-to-probability-statistics-and-random-processes/README.md rename to textbooks/001_introduction-to-probability-statistics-and-random-processes/README.md index d9878de..f854d9e 100644 --- a/study/001_introduction-to-probability-statistics-and-random-processes/README.md +++ b/textbooks/001_introduction-to-probability-statistics-and-random-processes/README.md @@ -4,7 +4,7 @@ total pages=1007 -**Currently reading:** chapter 3, page 222 +**Currently reading:** chapter 3, page 236 TODO: @@ -13,3 +13,4 @@ TODO: - 3.1.6 problems - 3.2.5 problems - ch3 end of chapter problems +- 4.1.4 problems diff --git a/study/001_introduction-to-probability-statistics-and-random-processes/ch1/end-of-chapter-problems.ipynb b/textbooks/001_introduction-to-probability-statistics-and-random-processes/ch1/end-of-chapter-problems.ipynb similarity index 100% rename from study/001_introduction-to-probability-statistics-and-random-processes/ch1/end-of-chapter-problems.ipynb rename to textbooks/001_introduction-to-probability-statistics-and-random-processes/ch1/end-of-chapter-problems.ipynb diff --git a/study/001_introduction-to-probability-statistics-and-random-processes/ch1/example-problems.ipynb b/textbooks/001_introduction-to-probability-statistics-and-random-processes/ch1/example-problems.ipynb similarity index 100% rename from study/001_introduction-to-probability-statistics-and-random-processes/ch1/example-problems.ipynb rename to textbooks/001_introduction-to-probability-statistics-and-random-processes/ch1/example-problems.ipynb diff --git a/study/001_introduction-to-probability-statistics-and-random-processes/ch1/notes.ipynb b/textbooks/001_introduction-to-probability-statistics-and-random-processes/ch1/notes.ipynb similarity index 100% rename from study/001_introduction-to-probability-statistics-and-random-processes/ch1/notes.ipynb rename to textbooks/001_introduction-to-probability-statistics-and-random-processes/ch1/notes.ipynb diff --git a/study/001_introduction-to-probability-statistics-and-random-processes/ch2/end-of-chapter-problems.ipynb b/textbooks/001_introduction-to-probability-statistics-and-random-processes/ch2/end-of-chapter-problems.ipynb similarity index 100% rename from study/001_introduction-to-probability-statistics-and-random-processes/ch2/end-of-chapter-problems.ipynb rename to 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a/study/001_introduction-to-probability-statistics-and-random-processes/ch4/end-of-chapter-problems.ipynb b/textbooks/001_introduction-to-probability-statistics-and-random-processes/ch4/end-of-chapter-problems.ipynb similarity index 100% rename from study/001_introduction-to-probability-statistics-and-random-processes/ch4/end-of-chapter-problems.ipynb rename to textbooks/001_introduction-to-probability-statistics-and-random-processes/ch4/end-of-chapter-problems.ipynb diff --git a/textbooks/001_introduction-to-probability-statistics-and-random-processes/ch4/example-problems.ipynb b/textbooks/001_introduction-to-probability-statistics-and-random-processes/ch4/example-problems.ipynb new file mode 100644 index 0000000..80c4aa2 --- /dev/null +++ b/textbooks/001_introduction-to-probability-statistics-and-random-processes/ch4/example-problems.ipynb @@ -0,0 +1,101 @@ +{ + "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 4 Example Problems" + ] + }, + { + "cell_type": "markdown", + "id": "5ce47b88", + "metadata": {}, + "source": [ + "4.2 \n", + "\n", + "a. \n", + "\n", + "$\\int_{-\\infty}^{\\infty}f_X(u)du = 1$\n", + "\n", + "ie\n", + "\n", + "\\begin{align*}\n", + "\\int_{-\\infty}^{0^+}f_X(u)du + \\int_{0}^{\\infty}f_X(x)dx = 1 \\\\\n", + "0 + \\int_{0}^{\\infty}ce^{-x}dx = 1 \\\\\n", + "c\\cdot\\lim_{t \\to \\infty }\\int_{0}^{t}e^{-x}dx = 1 \\\\\n", + "c\\cdot\\lim_{t \\to \\infty } [-e^{-x}]_0^t = 1 \\\\\n", + "c\\cdot\\lim_{t \\to \\infty } ((-e^{-t}) - (-e^{-0})) = 1 \\\\\n", + "c\\cdot\\lim_{t \\to \\infty } (-e^{-t} + 1) = 1 \\\\\n", + "c\\cdot 1 = 1 \\\\\n", + "c = 1\n", + "\\end{align*}\n", + "\n", + "b.\n", + "\n", + "Note that\n", + "\n", + "$$F_X(x) = \\int_{-\\infty}^x f_X(u)du$$\n", + "\n", + "and $c=1$\n", + "\n", + "so\n", + "\n", + "\\begin{align*}\n", + "F_X(x) = \\begin{cases} \n", + "1-e^{-x} & \\text{for } x \\geq 0 \\\\ \n", + "0 & \\text{otherwise} \n", + "\\end{cases}\n", + "\\end{align*}\n", + "\n", + "c. $F_X(3) - F_X(1) = $" + ] + }, + { + "cell_type": "markdown", + "id": "95a3b21c", + "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/textbooks/001_introduction-to-probability-statistics-and-random-processes/ch4/notes.ipynb b/textbooks/001_introduction-to-probability-statistics-and-random-processes/ch4/notes.ipynb new file mode 100644 index 0000000..cf63c34 --- /dev/null +++ b/textbooks/001_introduction-to-probability-statistics-and-random-processes/ch4/notes.ipynb @@ -0,0 +1,176 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 2, + "id": "c58309b2", + "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": "a6732353-51d5-4478-9cf8-5834e57e5a4e", + "metadata": {}, + "source": [ + "# Chapter 4 Notes" + ] + }, + { + "cell_type": "markdown", + "id": "9f0046c2", + "metadata": {}, + "source": [ + "## Continuous Random Variables and their Distributions (4.1.0)" + ] + }, + { + "cell_type": "markdown", + "id": "ea9b1f96", + "metadata": {}, + "source": [ + "***Definition*** A random variable $X$ with CDF $F_X(x)$ is said to be continuous if $F_X(x)$ is a continuous for all $x \\in \\mathbb{R}$" + ] + }, + { + "cell_type": "markdown", + "id": "a96da4d3", + "metadata": {}, + "source": [ + "## Probability Density Function (PDF) (4.1.1)" + ] + }, + { + "cell_type": "markdown", + "id": "c62129a5", + "metadata": {}, + "source": [ + "***Definition*** Consider a continuous random variable $X$ with an absolutely continuous CDF $F_X(x)$. The function $f_X(x)$ defined by\n", + "\n", + "$$f_X(x) = \\frac{dF_X(x)}{dx} = F'_X(x) \\quad \\text{if } F_X(x) \\text{ is differentiable at } x$$\n", + " \n", + "is called the probability density function (PDF) of $X$." + ] + }, + { + "cell_type": "markdown", + "id": "df411869", + "metadata": {}, + "source": [ + "NOTE: The PDF being constant implies uniformity\n", + "\n", + "NOTE: For small values of $\\delta$,\n", + "\n", + "$$P(x \\lt X \\leq x + \\delta) \\approx f_X(x)\\delta$$\n", + "\n", + "Thus if $f_X(x_1) \\gt f_X(x_2)$, we can say $P(x_1 \\lt X \\leq x_1 + \\delta) \\gt P(x_2 \\lt X \\leq x_2 + \\delta)$, ie the value of $X$ is more likely to be around $x_1$ then $x_2$\n", + "\n", + "NOTE: The CDF can be obtained from the PDF via (assuming absolute continuity)\n", + "\n", + "$$F_X(x) = \\int_{-\\infty}^x f_X(u)du$$" + ] + }, + { + "cell_type": "markdown", + "id": "f29a3bfb", + "metadata": {}, + "source": [ + "***Properties*** Consider a continuous random variable $X$ with PDF $f_X(x)$. We have\n", + "- $f_X(x) \\geq 0, \\forall x \\in \\mathbb{R}$\n", + "- $\\int_{-\\infty}^{\\infty}f_X(u)du = 1$\n", + "- $P(a \\lt X \\leq b) = F_X(b) - F_X(a) = \\int_a^bf_X(u)du$\n", + "- For a set $A$, $P(X \\in A) = \\int_Af_X(u)du$. However, set $A$ must satisfy:" + ] + }, + { + "cell_type": "markdown", + "id": "6f546fed", + "metadata": {}, + "source": [ + "***Definition*** If $X$ is a continuous random variable, we can write the range of $X$ as\n", + "\n", + "$$R_X = \\{ x \\mid f_X(x) \\gt 0 \\}$$" + ] + }, + { + "cell_type": "markdown", + "id": "170db3a0", + "metadata": {}, + "source": [ + "***Property*** The expected value if a continuous random variable $X$ is\n", + "\n", + "$$E[X] = \\int_{-\\infty}^{\\infty}xf_X(x)dx$$" + ] + }, + { + "cell_type": "markdown", + "id": "deccea51", + "metadata": {}, + "source": [ + "***Property*** Law of the unconscious statistician (LOTUS) for continuous random variables\n", + "\n", + "$$E[g(X)] = \\int_{-\\infty}^{\\infty}g(x)f_X(x)dx$$\n" + ] + }, + { + "cell_type": "markdown", + "id": "c0a20195", + "metadata": {}, + "source": [ + "***Property*** Variance for a continuous random variable $X$, we can write\n", + "\n", + "\\begin{align*}\n", + "\\text{Var}(X) &= E[(X - E[x])^2] = \\int_{-\\infty}^{\\infty}(x - E[X])^2f_X(x)dx \\\\\n", + "&= E[X^2] - E[X]^2 = \\int_{-\\infty}^{\\infty}x^2f_X(x)dx - E[X]^2\n", + "\\end{align*}" + ] + }, + { + "cell_type": "markdown", + "id": "6f2f6b72", + "metadata": {}, + "source": [ + "## Functions of Continuous Random Variables" + ] + }, + { + "cell_type": "markdown", + "id": "f4a95738", + "metadata": {}, + "source": [ + "***Theorem***" + ] + } + ], + "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/public/conditional_prob_tree.png b/textbooks/001_introduction-to-probability-statistics-and-random-processes/public/conditional_prob_tree.png similarity index 100% rename from study/001_introduction-to-probability-statistics-and-random-processes/public/conditional_prob_tree.png rename to textbooks/001_introduction-to-probability-statistics-and-random-processes/public/conditional_prob_tree.png diff --git a/study/001_introduction-to-probability-statistics-and-random-processes/textbooks/Introduction to Probability, Statistics, and Random Processes - 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