From 3959b06668c93a75ea36ad5ede55eca9c03fb8e5 Mon Sep 17 00:00:00 2001 From: Caleb Burke Date: Mon, 25 May 2026 18:03:02 -0700 Subject: [PATCH] chapter 1 progress --- .../README.md | 2 +- .../ch1/notes.ipynb | 604 ++++++++++++++++++ .../ch1/summary.ipynb | 131 ++++ .../notebooks/ch1.ipynb | 488 -------------- .../public/conditional_prob_tree.png | Bin 0 -> 35829 bytes 5 files changed, 736 insertions(+), 489 deletions(-) create mode 100644 study/001_introduction-to-probability-statistics-and-random-processes/ch1/notes.ipynb create mode 100644 study/001_introduction-to-probability-statistics-and-random-processes/ch1/summary.ipynb delete mode 100644 study/001_introduction-to-probability-statistics-and-random-processes/notebooks/ch1.ipynb create mode 100644 study/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/README.md b/study/001_introduction-to-probability-statistics-and-random-processes/README.md index 2e6aca5..b57b9f3 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 @@ -2,4 +2,4 @@ total pages=1007 -**Currently reading:** chapter 1, page 56 +**Currently reading:** chapter 1, page 73 diff --git a/study/001_introduction-to-probability-statistics-and-random-processes/ch1/notes.ipynb b/study/001_introduction-to-probability-statistics-and-random-processes/ch1/notes.ipynb new file mode 100644 index 0000000..2314251 --- /dev/null +++ b/study/001_introduction-to-probability-statistics-and-random-processes/ch1/notes.ipynb @@ -0,0 +1,604 @@ +{ + "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 1 Notes" + ] + }, + { + "cell_type": "markdown", + "id": "48d9ec9e-83da-40ca-ae79-3c45f8af137c", + "metadata": {}, + "source": [ + "## Main Concepts\n", + "\n", + "Outcome: A result of a random experiment.\n", + "\n", + "Sample Space: The set of all possible outcomes.\n", + "\n", + "Event: A subset of the sample space.\n", + "\n", + "Inclusion-exclusion principle holds for probability\n", + "\n", + "Consider a sample space S. If S is a countable set, this refers to a discrete probability\n", + "mode\n" + ] + }, + { + "cell_type": "markdown", + "id": "7ac122be-50b2-423c-b88f-e4b3327b21bd", + "metadata": {}, + "source": [ + "## Example Problems" + ] + }, + { + "cell_type": "markdown", + "id": "7fb87a35-a470-4d98-935f-80c814e3f95d", + "metadata": {}, + "source": [ + "Example 1.5 - soln\n", + "\n", + "- there are 10 people with white shirts and 8 people with red shirts;\n", + "- 4 people have black shoes and white shirts\n", + "- 3 people have black shoes and red shirts\n", + "- the total number of people with white or red shirts or black shoes is 21\n", + "\n", + "Let A be the set of people with white shirts, B be the set of people with red shirts and let C be the set of people with black shoes.\n", + "\n", + "\\begin{align*}\n", + "|A|=10 \\\\\n", + "|B|=8 \\\\\n", + "|A \\cap C| = 4 \\\\\n", + "|B \\cap C| = 3 \\\\\n", + "|A \\cup B \\cup C| = 21\n", + "\\end{align*}\n", + "\n", + "Now we solve for $|C|$:\n", + "\n", + "\\begin{align*}\n", + "|A| + |B| + |C| - |A \\cap B| - |A \\cap C| - |B \\cap C| + |A \\cap B \\cap C| = 21 \\\\\n", + "10 + 8 + |C| - 0 - 4 - 3 - 0 = 21 \\\\\n", + "18 + |C| - 7 = 21 \\\\\n", + "|C| + 11 = 21 \\\\\n", + "|C| = 10\n", + "\\end{align*}\n", + "\n", + "$\\therefore$ number of people with black shoes is 10\n" + ] + }, + { + "cell_type": "markdown", + "id": "72b40733-531a-48f3-9879-75601684afc2", + "metadata": {}, + "source": [ + "Example 1.11 - soln\n", + "\n", + "Suppose we have the following information:\n", + "1. There is a 60 percent chance that it will rain today.\n", + "2. There is a 50 percent chance that it will rain tomorrow.\n", + "3. There is a 30 percent chance that it does not rain either day.\n", + "\n", + "T = rains\n", + "F = no rain\n", + "\n", + "$S = \\{(F, F), (F, T), (T, F), (T, T)\\}$\n", + "\n", + "$P((T, F) \\cup (T, T)) = 0.6$\n", + "\n", + "$P((F, T) \\cup (T, T)) = 0.5$\n", + "\n", + "$P((F, F)) = 0.3$\n", + "\n", + "\\begin{align*}\n", + "P(S) = 1 \\\\\n", + "P(\\{(F, F)\\} \\cup \\{(F, T)\\} \\cup \\{(T, F)\\} \\cup \\{(T, T)\\}) = 1 \\\\\n", + "P((F,F)) + P(\\{(F, T)\\} \\cup \\{(T, F)\\} \\cup \\{(T, T)\\}) = 1 \\\\\n", + "0.3 + P(\\{(F, T)\\} \\cup \\{(T, F)\\} \\cup \\{(T, T)\\}) = 1 \\\\\n", + "P(\\{(F, T)\\} \\cup \\{(T, F)\\} \\cup \\{(T, T)\\}) = 0.7 \\\\\n", + "P(\\{(F, T)\\} \\cup \\{(T, T)\\}) + P((T, F)) = 0.7 \\\\\n", + "0.5 + P((T, F)) = 0.7 \\\\\n", + "P((T, F)) = 0.2 \\\\\n", + "P(\\{(T, F)\\} \\cup \\{(T, T)\\}) + P((F, T)) = 0.7 \\\\\n", + "P((F, T)) = 0.1\n", + "\\end{align*}\n", + "\n", + "Find the following probabilities:\n", + "\n", + "a. The probability that it will rain today or tomorrow.\n", + "\n", + "\\begin{align*}\n", + "P((T, F) \\cup (F, T) \\cup (T, T)) = 0.7\n", + "\\end{align*}\n", + "\n", + "b. The probability that it will rain today and tomorrow.\n", + "\n", + "\\begin{align*}\n", + "P((T, T)) = 1 - 0.3 - 0.2 - 0.1 = 0.4\n", + "\\end{align*}\n", + "\n", + "c. The probability that it will rain today but not tomorrow.\n", + "\n", + "\\begin{align*}\n", + "P((T, F)) = 0.2\n", + "\\end{align*}\n", + "\n", + "d. The probability that it either will rain today or tomorrow, but not both.\n", + "\n", + "\\begin{align*} \n", + "P(\\{(T, F)\\} \\cup \\{(F, T)\\}) = P((T, F)) + P((F, T)) = 0.2 + 0.1 = 0.3\n", + "\\end{align*}\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "id": "8b5131dd-5ebd-4156-b808-f8df273317fb", + "metadata": {}, + "source": [ + "Example 1.12 - soln\n", + "\n", + "$S = \\{ -1, 0, 1, 2, 3, ... \\}$\n", + "\n", + "$\\forall x \\in S, P(x) = \\frac{1}{2^{x + 2}}$\n", + "\n", + "What is the probability that I win more than or equal to 1 dollar and less than 4 dollars?\n", + "\n", + "\\begin{align*} \n", + "P({1, 2, 3}) = P(1) + P(2) + P(3) \\\\\n", + "= 1/8 + 1/16 + 1/32\n", + "\\end{align*}\n", + "\n", + "What is the probability that I win more than 2 dollars?\n", + "\n", + "\\begin{align*} \n", + "\\sum_{i=3}^{\\infty} P(i) = P(3) + P(4) + P(5) + P(6) + ... \\\\\n", + "= 1/32 + 1/64 + 1/128 + 1/256 + ... \\\\\n", + "=\\frac{\\frac{1}{32}}{1 - \\frac{1}{2}}\n", + "=\\frac{1}{16}\n", + "\\end{align*}" + ] + }, + { + "cell_type": "markdown", + "id": "4bc9a64b", + "metadata": {}, + "source": [ + "# 1.3.6 Problems. \n", + "\n", + "## Problem 1 - skipping\n", + "\n", + "## Problem 2 \n", + "\n", + "a. \n", + "\n", + "\\begin{align*}\n", + "S = \\{ x \\mid x >= 2, x \\in \\mathbb{N} \\}\n", + "\\end{align*}\n", + "\n", + "b.\n", + "1 red, 1 blue, 1 white and 1 green\n", + "\n", + "\\begin{align*}\n", + "S = \\{ (R, B), (B, R), (R, W), (W, R), (R, G), (G, R), (B, W), (W, B), (B, G), (G, B), (W, G), (G, W) \\}\n", + "\\end{align*}\n", + "\n", + "c.\n", + "\n", + "$S = [0, \\frac{1}{3})$\n", + "\n", + "## Problem 3\n", + "\n", + "- $A \\cup B \\cup C = S$\n", + "- $P(A) = \\frac{3}{6}$\n", + "- $P(B) = \\frac{4}{6}$\n", + "- $P(A \\cup B) = \\frac{5}{6}$\n", + "\n", + "### a. Find $P(A \\cap B)$\n", + "\n", + "$P(A \\cap B) = P(A) + P(B) - P(A \\cup B) = \\frac{3}{6} + \\frac{4}{6} - \\frac{5}{6} = \\frac{2}{6}$\n", + "\n", + "### b. Do $A$, $B$ and $C$ form a partition of $S$?\n", + "\n", + "No. Proof by contridiction:\n", + "\n", + "Assume $A$, $B$ and $C$ form a partition of $S$. That would imply $P(S) = P(A) + P(B) + P(C) = 1$.\n", + "\n", + "$P(A) = \\frac{3}{6}$ and $P(B) = \\frac{4}{6}$, so $P(A) + P(B) = \\frac{7}{6}$\n", + "\n", + "$P(A) + P(B) > P(S)$ therefore contirdiction.\n", + "\n", + "$A$, $B$ and $C$ do not form a partition of $S$. $\\blacksquare$\n", + "\n", + "### c. Find $P(C \\setminus (A \\cup B))$\n", + "\n", + "\\begin{align*}\n", + "A \\cup B \\cup C = S \\\\\n", + "A \\cup B \\cup C \\setminus (A \\cup B) = S \\setminus (A \\cup B) \\\\\n", + "C \\setminus (A \\cup B) = S \\setminus (A \\cup B)\n", + "\\end{align*}\n", + "\n", + "Therefore $P(C \\setminus (A \\cup B)) = P(S \\setminus (A \\cup B))$.\n", + "\n", + "Notice $P(S \\setminus (A \\cup B)) = P((A \\cup B)^c)$, so $P(C \\setminus (A \\cup B)) = P((A \\cup B)^c)$\n", + "\n", + "So\n", + "\n", + "\\begin{align*}\n", + "P(C - (A \\cup B)) &= P((A \\cup B)^c) \\\\\n", + "&= P(S) - P(A \\cup B) \\\\\n", + "&= 1 - \\frac{5}{6} \\\\\n", + "&= \\frac{1}{6}\n", + "\\end{align*}\n", + "\n", + "### d. If $P(C \\cap (A \\cup B)) = \\frac{5}{12}, P(C) =$ ?\n", + "\n", + "$C = (C \\cap (A \\cup B)) \\cup C \\setminus (A \\cup B)$\n", + "\n", + "Therefore\n", + "\n", + "\\begin{align*}\n", + "P(C) &= P((C \\cap (A \\cup B)) \\cup C \\setminus (A \\cup B)) \\\\\n", + "&= P(C \\cap (A \\cup B)) + P(C \\setminus (A \\cup B)) \\\\\n", + "&= \\frac{5}{12} + \\frac{1}{6} \\\\\n", + "&= \\frac{7}{12}\n", + "\\end{align*}\n", + "\n", + "## Problem 4.\n", + "\n", + "### a. \n", + "Let $X$ be our sample space for the first row, and $Y$ the second roll. ($X \\cup Y = S$)\n", + "\n", + "$X = \\{ 1, 2, 3, 4, 5, 6 \\}$\n", + "\n", + "Let $x$ be the first dice row and $y$ the second dice roll. ($x \\in X$ and $y \\in Y$)\n", + "\n", + "The probability $x < y$ is $|X| - x$ or $6 - x$.\n", + "\n", + "So \n", + "\n", + "\\begin{align*}\n", + "\\sum_{x \\in X} (6 - x) &= \\sum_{x \\in X} 6 - \\sum_{x \\in X} x \\\\\n", + "&= 36 - (1 + 2 + 3 + 4 + 5 + 6) \\\\\n", + "&= 15\n", + "\\end{align*}\n", + "\n", + "Therefore $P(A) = \\frac{15}{36}$\n", + "\n", + "### b.\n", + "\n", + "Number of outcomes where 6 is only first: 1 * 5\n", + "Number of outcomes where 6 is only second: 1 * 5 \n", + "Number of outcomes where 6 is both: 1\n", + "\n", + "Therefore $P(B) = \\frac{11}{36}$\n", + "\n", + "## Problem 5." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "e589ecc1", + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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", + "text/plain": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "2.25091334378745\n" + ] + } + ], + "source": [ + "xmin = 0\n", + "xmax = 20\n", + "\n", + "x = np.linspace(xmin, xmax, 200)\n", + "y = 1 - np.exp((-1 * (x / 5)))\n", + "\n", + "# ----------------------------------------------------\n", + "\n", + "plt.figure(figsize=(8, 5))\n", + "\n", + "# 1. Updated the LaTeX label to match your new math\n", + "plt.plot(x, y, color=\"crimson\", linewidth=2.5, label=r\"$y = e^{-x/5}$\")\n", + "\n", + "plt.xlim(xmin, xmax)\n", + "\n", + "# 2. Tightened the Y-limits so you can actually see the decay curve\n", + "plt.ylim(0, 1)\n", + "\n", + "plt.legend(fontsize=12, loc=\"upper right\") # Moved to upper right since curve decays downward\n", + "plt.show()\n", + "\n", + "# print((1 - 2.71 ** (-3/5)) * 5)" + ] + }, + { + "cell_type": "markdown", + "id": "00afcbd1", + "metadata": {}, + "source": [ + "### a.\n", + "\n", + "Our sample space $S$ is the time at which the product breaks. The product could break immediatly ($T = 0$) or never break ($T \\to \\infty$)\n", + "\n", + "Therefore $S = [0, \\infty)$\n", + "\n", + "### b.\n", + "\n", + "$P(T \\geq 0) = e^{-\\frac{t}{5}} = 1$\n", + "\n", + "This makes sense, as this states that the probability the product doesnt break immediatly is 1. This would have to be the case, otherwise the product was broken to begin with, which is an impossible senario.\n", + "\n", + "$\\lim_{t \\to \\infty} P(T \\geq t) = \\lim_{t \\to \\infty} e^{-\\frac{t}{5}} = 0$ (skiping proof, can be easily seen in graph above)\n", + "\n", + "This means the as time goes on, the probability the product doesnt break approachs 0. This would imply the product will eventually break given infinite time.\n", + "\n", + "### c.\n", + "\n", + "$t_1 \\lt t_2 \\implies P(T \\geq t_1) \\geq P(T \\geq t_2)$.\n", + "\n", + "Proof\n", + "\n", + "\\begin{align*}\n", + "t_1 \\leq t_2 \\\\\n", + "-\\frac{t_1}{5} \\cdot -5 \\geq -\\frac{t_2}{5} \\cdot -5 \\\\\n", + "-\\frac{t_1}{5} \\geq -\\frac{t_2}{5} \\\\\n", + "e^{-\\frac{t_1}{5}} \\geq e^{-\\frac{t_2}{5}} \\\\\n", + "P(T \\geq t_1) \\geq P(T \\geq t_2) \n", + "\\end{align*}\n", + "\n", + "$\\blacksquare$\n", + "\n", + "This implies a more time means more probability the product may breakdown which makes intuitive sense.\n", + "\n", + "### d. Find the probability that the product breaks down within three years of the purchase time.\n", + "\n", + "\n", + "\n", + "### e. Find the probability that the product breaks down in the second year\n", + "\n", + "$P(1 \\leq T \\lt 2)=$?\n", + "\n", + "\\begin{align*}\n", + "P(1 \\leq T \\lt 2)=P(T \\geq 1) + P(T \\lt 2)\n", + "\\end{align*}\n", + "\n", + "## Problem 6\n", + "\n", + "Triangle inequaility:\n", + "\n", + "$x + y \\gt z$\n", + "\n", + "$y + z \\gt x$\n", + "\n", + "$x + z \\gt y$\n", + "\n", + "Since $x + y + z = 1$ then:\n", + "\n", + "$x \\lt \\frac{1}{2}$\n", + "\n", + "$y \\lt \\frac{1}{2}$\n", + "\n", + "$z \\lt \\frac{1}{2}$\n", + "\n", + "The area of our sample space is the plane where $x + y + z = 1$ in $\\mathbb{R}^3$, which is the triangle with points $(1, 0, 0)$, $(0, 1, 0)$, and $(0, 0, 1)$.\n", + "\n", + "Area of $S$ is $\\frac{\\sqrt{3}}{2}$ (skipping proof).\n", + "\n", + "The triangle $A$ is $\\frac{\\sqrt{3}}{8}$\n", + "\n", + "$P(A) = \\frac{|A|}{|S|} = \\frac{\\frac{\\sqrt{3}}{8}}{\\frac{\\sqrt{3}}{2}} = \\frac{\\sqrt{3}}{8} \\cdot \\frac{2}{\\sqrt{3}} = \\frac{1}{4}$\n" + ] + }, + { + "cell_type": "markdown", + "id": "4609aba5-dfdc-4500-bec2-cfa2e544cca7", + "metadata": {}, + "source": [ + "\n", + "### Example 1.17 - soln\n", + "\n", + "I roll a fair die twice and obtain two numbers X1 = result of the first roll and X2 = result\n", + "of the second roll. Given that I know X1 + X2 = 7, what is the probability that X1 = 4 or\n", + "X2 = 4?\n", + "\n", + "What is our sample space?\n", + "\n", + "\\begin{align*} \n", + "D = \\{1,2,3,4,5,6\\} \\\\\n", + "S = D \\times D\n", + "\\end{align*}\n", + "\n", + "\\begin{align*} \n", + "P(A|B) = \\frac{P(A \\cap B)}{P(B)}\n", + "\\end{align*}\n", + "\n", + "\\begin{align*} \n", + "A = \\{(x,y) \\mid x=4 \\lor y = 4 \\} \\\\\n", + "|A| = 12 \\\\\n", + "P(A) = \\frac{|A|}{|S|} \\\\\n", + "= \\frac{12}{36} = \\frac{1}{3}\n", + "\\end{align*}\n", + "\n", + "> NOTE my event set A may be overcounting \n", + "\n", + "\\begin{align*} \n", + "B = \\{(x,y) \\mid x+y = 7 , x,y \\in \\mathbb{N}\\} \\\\\n", + "|B| = 6 \\\\\n", + "P(B) = \\frac{1}{6} \\\\\n", + "\\end{align*}\n", + "\n", + "\\begin{align*} \n", + "P(A \\cap B) = \\frac{1}{18}\n", + "\\end{align*}\n", + "\n", + "\n", + "\\begin{align*} \n", + "B = \\{(x,y) \\mid x+y = 7 , x,y \\in \\mathbb{N}\\} \\\\\n", + "A = \\{(x,y) \\mid x=4 \\lor y = 4 \\} \\\\\n", + "P(A|B) = \\frac{P(A \\cap B)}{P(B)} \\\\\n", + "= \\frac{\\frac{1}{18} }{ \\frac{1}{6} } \\\\\n", + "= \\frac{6}{18} = \\frac{1}{3}\n", + "\\end{align*}\n" + ] + }, + { + "cell_type": "markdown", + "id": "12c8a0d4", + "metadata": {}, + "source": [ + "## Example 1.18\n", + "\n", + "### a.\n", + "\n", + "$S_\\text{conditional} = \\{\\{G, G\\}, \\{G, B\\}\\}$, therefore $\\frac{1}{2}$\n", + "\n", + "### b.\n", + "\n", + "$S_\\text{conditional} = \\{\\{G, G\\}, \\{G, B\\}, \\{B, G\\}\\}$, therefore $\\frac{1}{3}$" + ] + }, + { + "cell_type": "markdown", + "id": "1305f9d6", + "metadata": {}, + "source": [ + "$$P(B,C \\mid A) = P(B \\mid A) \\cdot P(C \\mid A,B)$$\n", + "\n", + "$$P(A_1 \\cap A_2 \\cap \\dots \\cap A_n \\mid E) = P(A_1 \\mid E) \\cdot P(A_2 \\mid A_1, E) \\cdot P(A_3 \\mid A_1, A_2, E) \\cdots P(A_n \\mid A_1, \\dots, A_{n-1}, E)$$" + ] + }, + { + "cell_type": "markdown", + "id": "ca15c14a", + "metadata": {}, + "source": [ + "### Example 1.19\n", + "\n", + "In a factory there are 100 units of a certain product, 5 of which are defective. We pick\n", + "three units from the 100 units at random. What is the probability that none of them are\n", + "defective?\n", + "\n", + "Let $A_1$ be the event where we pick a non defective unit first, $A_2$ on our second, and $A_3$ on our third.\n", + "\n", + "We want to find $P(A_1 \\cap A_2 \\cap A_3)$.\n", + "\n", + "Note that $P(A_1 \\cap A_2 \\cap A_3) = P(A_1)P(A_2|A_1)P(A_3|A_1,A_2)$\n", + "\n", + "$$P(A_1) = \\frac{95}{100}$$\n", + "\n", + "$$P(A_2|A_1) = \\frac{94}{99}$$\n", + "\n", + "$$P(A_3|A_1,A_2) = \\frac{93}{98}$$\n", + "\n", + "\\begin{align*} \n", + "P(A)P(B|A)P(C|A,B) &= \\frac{95}{100} \\cdot \\frac{94}{99} \\cdot \\frac{93}{98} \\\\\n", + "\\approx 0.856\n", + "\\end{align*}" + ] + }, + { + "cell_type": "markdown", + "id": "4790a1dc", + "metadata": {}, + "source": [ + "## Example 1.20\n", + "\n", + "$S = [1:11]$\n", + "\n", + "Let $A$ be the event $N \\lt 7$, $N \\in S$\n", + "\n", + "Let $B$ be the event $N$ is even. Are $A$ and $B$ independent?\n", + "\n", + "$$P(A) = \\frac{|A|}{|S|} = \\frac{6}{10} = \\frac{3}{5}$$\n", + "\n", + "Note that $|B| = 5$.\n", + "\n", + "$$P(A|B) = \\frac{|A \\cap B|}{|B|} = \\frac{3}{5}$$\n", + "\n", + "$P(A|B) = P(A) \\therefore A$ and $B$ are independent. \n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "ef40221f", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.8559987631416203" + ] + }, + "execution_count": 1, + "metadata": {}, + "output_type": "execute_result" + } + ], + "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/ch1/summary.ipynb b/study/001_introduction-to-probability-statistics-and-random-processes/ch1/summary.ipynb new file mode 100644 index 0000000..2bb6144 --- /dev/null +++ b/study/001_introduction-to-probability-statistics-and-random-processes/ch1/summary.ipynb @@ -0,0 +1,131 @@ +{ + "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": [ + "# Summary of Introduction to Probability, Statistics, and Random Processes (Hossein Pishro-Nik) chapter 1" + ] + }, + { + "cell_type": "markdown", + "id": "be70f5df", + "metadata": {}, + "source": [ + "## Intro (1.0.0 - 1.3.1)\n", + "\n", + "**Theorem 1.1: De Morgan's law** \\\n", + "For any sets $A_1, A_2, A_3, \\dots A_n$, we have\n", + "- $(A_1 \\cup A_2 \\cup A_3 \\cup \\dots A_n)^c = A_1^c \\cap A_2^c \\cap A_3^c \\cap \\dots A_n^c$\n", + "- $(A_1 \\cap A_2 \\cap A_3 \\cap \\dots A_n)^c = A_1^c \\cup A_2^c \\cup A_3^c \\cup \\dots A_n^c$\n", + "\n", + "**Theorem 1.2: Distributive law** \\\n", + "For any sets $A, B,$ and $C$ we have\n", + "- $A \\cap (B \\cup C) = (A \\cap B)\\cup(A \\cap C)$\n", + "- $A \\cup (B \\cap C) = (A \\cup B)\\cap(A \\cup C)$\n", + "\n", + "**Inclusion-exclusion principle** \\\n", + "For a finite collection of sets $A_1, A_2, A_3, \\dots A_n$, we have\n", + "\n", + "$\\left| \\bigcup_{i=1}^n A_i \\right| = \\sum_{i=1}^n |A_i| - \\sum_{i < j} |A_i \\cap A_j| + \\sum_{i < j < k} |A_i \\cap A_j \\cap A_k| - \\dots + (-1)^{n-1} |A_1 \\cap A_2 \\cap \\dots \\cap A_n|$\n", + "\n", + "$n = 2$ case:\n", + "\n", + "$|A \\cup B| = |A| + |B| - |A \\cap B|$\n", + "\n", + "$n = 3$ case:\n", + "\n", + "$|A \\cup B \\cup C| = |A| + |B| + |C| - |A \\cap B| - |A \\cap C| - |B \\cap C| + |A \\cap B \\cap C|$\n", + "\n", + "## Random experiments (1.3.1 - 1.4)\n", + "\n", + "- A **random experiment** is a process by which we observe something uncertain\n", + "- An **outcome** is a result of a random experiment\n", + "- The **sample space** $S$ is the set of all possible outcomes\n", + "- An **event** $A$ is any subset of $S$\n", + "\n", + "> In the context of a random experiment, the sample space is our *universal set*\n", + "\n", + "**Axioms of Probability**\n", + "\n", + "1. For any event $A$, $P(A) \\geq 0$\n", + "2. $P(S) = 1$\n", + "3. If $A_1, A_2, A_3, \\dots$ are disjoint events, then $P(A_1 \\cup A_2 \\cup A_3 \\cup \\dots) = P(A_1) + P(A_2) + P(A_3) + \\dots$\n", + "\n", + "**Some notation**\n", + "\n", + "- $P(A \\cap B) = P(A$ and $B) = P(A,B)$\n", + "- $P(A \\cup B) = P(A$ or $B)$\n", + "\n", + "In a finite sample space $S$, where all outcomes are equally likely, the probability of any event $A$ can be found by\n", + "\n", + "\\begin{align*}\n", + "P(A) = \\frac{|A|}{|S|}\n", + "\\end{align*}\n", + "\n", + "## Conditional probability (1.4.0)\n", + "\n", + "If $A$ and $B$ are twos events in sample space $S$, then the **conditional probability of $A$ given $B$** is defined as\n", + "\n", + "\\begin{align*}\n", + "P(A|B) = \\frac{|A \\cap B|}{|B|}, \\text{when } P(B) > 0\n", + "\\end{align*}\n", + "\n", + "For events $A, B,$ and $C$, with $P(C) \\gt 0$, we have\n", + "\n", + "- $P(A^c|C) = 1 - P(A|C)$\n", + "- $P(\\empty|C) = 0$\n", + "- $P(A|C) \\leq 1$\n", + "- $P(A \\setminus B|C) = P(A|C) - P(A \\cap B|C)$\n", + "- $P(A \\cup B|C) = P(A|C) + P(B|C) - P(A \\cap B|C)$\n", + "- if $A \\subset B$ then $P(A|C) \\leq P(B|C)$\n", + "\n", + "![](./public/conditional_prob_tree.png)\n", + "\n", + "## Independence (1.4.1)\n", + "\n", + "**Definition.** Two events $A$ and $B$ are *independent* if $P(A \\cap B) = P(A)P(B)$. AKA $P(A|B) = P(A)$" + ] + } + ], + "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/notebooks/ch1.ipynb b/study/001_introduction-to-probability-statistics-and-random-processes/notebooks/ch1.ipynb deleted file mode 100644 index 098518e..0000000 --- a/study/001_introduction-to-probability-statistics-and-random-processes/notebooks/ch1.ipynb +++ /dev/null @@ -1,488 +0,0 @@ -{ - "cells": [ - { - "cell_type": "code", - "execution_count": 9, - "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 1 Notes" - ] - }, - { - "cell_type": "markdown", - "id": "48d9ec9e-83da-40ca-ae79-3c45f8af137c", - "metadata": {}, - "source": [ - "## Main Concepts\n", - "\n", - "Outcome: A result of a random experiment.\n", - "\n", - "Sample Space: The set of all possible outcomes.\n", - "\n", - "Event: A subset of the sample space.\n", - "\n", - "Inclusion-exclusion principle holds for probability\n", - "\n", - "Consider a sample space S. If S is a countable set, this refers to a discrete probability\n", - "mode\n" - ] - }, - { - "cell_type": "markdown", - "id": "7ac122be-50b2-423c-b88f-e4b3327b21bd", - "metadata": {}, - "source": [ - "## Example Problems" - ] - }, - { - "cell_type": "markdown", - "id": "7fb87a35-a470-4d98-935f-80c814e3f95d", - "metadata": {}, - "source": [ - "Example 1.5 - soln\n", - "\n", - "- there are 10 people with white shirts and 8 people with red shirts;\n", - "- 4 people have black shoes and white shirts\n", - "- 3 people have black shoes and red shirts\n", - "- the total number of people with white or red shirts or black shoes is 21\n", - "\n", - "Let A be the set of people with white shirts, B be the set of people with red shirts and let C be the set of people with black shoes.\n", - "\n", - "\\begin{align*}\n", - "|A|=10 \\\\\n", - "|B|=8 \\\\\n", - "|A \\cap C| = 4 \\\\\n", - "|B \\cap C| = 3 \\\\\n", - "|A \\cup B \\cup C| = 21\n", - "\\end{align*}\n", - "\n", - "Now we solve for $|C|$:\n", - "\n", - "\\begin{align*}\n", - "|A| + |B| + |C| - |A \\cap B| - |A \\cap C| - |B \\cap C| + |A \\cap B \\cap C| = 21 \\\\\n", - "10 + 8 + |C| - 0 - 4 - 3 - 0 = 21 \\\\\n", - "18 + |C| - 7 = 21 \\\\\n", - "|C| + 11 = 21 \\\\\n", - "|C| = 10\n", - "\\end{align*}\n", - "\n", - "$\\therefore$ number of people with black shoes is 10\n" - ] - }, - { - "cell_type": "markdown", - "id": "72b40733-531a-48f3-9879-75601684afc2", - "metadata": {}, - "source": [ - "Example 1.11 - soln\n", - "\n", - "Suppose we have the following information:\n", - "1. There is a 60 percent chance that it will rain today.\n", - "2. There is a 50 percent chance that it will rain tomorrow.\n", - "3. There is a 30 percent chance that it does not rain either day.\n", - "\n", - "T = rains\n", - "F = no rain\n", - "\n", - "$S = \\{(F, F), (F, T), (T, F), (T, T)\\}$\n", - "\n", - "$P((T, F) \\cup (T, T)) = 0.6$\n", - "\n", - "$P((F, T) \\cup (T, T)) = 0.5$\n", - "\n", - "$P((F, F)) = 0.3$\n", - "\n", - "\\begin{align*}\n", - "P(S) = 1 \\\\\n", - "P(\\{(F, F)\\} \\cup \\{(F, T)\\} \\cup \\{(T, F)\\} \\cup \\{(T, T)\\}) = 1 \\\\\n", - "P((F,F)) + P(\\{(F, T)\\} \\cup \\{(T, F)\\} \\cup \\{(T, T)\\}) = 1 \\\\\n", - "0.3 + P(\\{(F, T)\\} \\cup \\{(T, F)\\} \\cup \\{(T, T)\\}) = 1 \\\\\n", - "P(\\{(F, T)\\} \\cup \\{(T, F)\\} \\cup \\{(T, T)\\}) = 0.7 \\\\\n", - "P(\\{(F, T)\\} \\cup \\{(T, T)\\}) + P((T, F)) = 0.7 \\\\\n", - "0.5 + P((T, F)) = 0.7 \\\\\n", - "P((T, F)) = 0.2 \\\\\n", - "P(\\{(T, F)\\} \\cup \\{(T, T)\\}) + P((F, T)) = 0.7 \\\\\n", - "P((F, T)) = 0.1\n", - "\\end{align*}\n", - "\n", - "Find the following probabilities:\n", - "\n", - "a. The probability that it will rain today or tomorrow.\n", - "\n", - "\\begin{align*}\n", - "P((T, F) \\cup (F, T) \\cup (T, T)) = 0.7\n", - "\\end{align*}\n", - "\n", - "b. The probability that it will rain today and tomorrow.\n", - "\n", - "\\begin{align*}\n", - "P((T, T)) = 1 - 0.3 - 0.2 - 0.1 = 0.4\n", - "\\end{align*}\n", - "\n", - "c. The probability that it will rain today but not tomorrow.\n", - "\n", - "\\begin{align*}\n", - "P((T, F)) = 0.2\n", - "\\end{align*}\n", - "\n", - "d. The probability that it either will rain today or tomorrow, but not both.\n", - "\n", - "\\begin{align*} \n", - "P(\\{(T, F)\\} \\cup \\{(F, T)\\}) = P((T, F)) + P((F, T)) = 0.2 + 0.1 = 0.3\n", - "\\end{align*}\n", - "\n" - ] - }, - { - "cell_type": "markdown", - "id": "8b5131dd-5ebd-4156-b808-f8df273317fb", - "metadata": {}, - "source": [ - "Example 1.12 - soln\n", - "\n", - "$S = \\{ -1, 0, 1, 2, 3, ... \\}$\n", - "\n", - "$\\forall x \\in S, P(x) = \\frac{1}{2^{x + 2}}$\n", - "\n", - "What is the probability that I win more than or equal to 1 dollar and less than 4 dollars?\n", - "\n", - "\\begin{align*} \n", - "P({1, 2, 3}) = P(1) + P(2) + P(3) \\\\\n", - "= 1/8 + 1/16 + 1/32\n", - "\\end{align*}\n", - "\n", - "What is the probability that I win more than 2 dollars?\n", - "\n", - "\\begin{align*} \n", - "\\sum_{i=3}^{\\infty} P(i) = P(3) + P(4) + P(5) + P(6) + ... \\\\\n", - "= 1/32 + 1/64 + 1/128 + 1/256 + ... \\\\\n", - "=\\frac{\\frac{1}{32}}{1 - \\frac{1}{2}}\n", - "=\\frac{1}{16}\n", - "\\end{align*}" - ] - }, - { - "cell_type": "markdown", - "id": "4bc9a64b", - "metadata": {}, - "source": [ - "# 1.3.6 Problems. \n", - "\n", - "## Problem 1 - skipping\n", - "\n", - "## Problem 2 \n", - "\n", - "a. \n", - "\n", - "\\begin{align*}\n", - "S = \\{ x \\mid x >= 2, x \\in \\mathbb{N} \\}\n", - "\\end{align*}\n", - "\n", - "b.\n", - "1 red, 1 blue, 1 white and 1 green\n", - "\n", - "\\begin{align*}\n", - "S = \\{ (R, B), (B, R), (R, W), (W, R), (R, G), (G, R), (B, W), (W, B), (B, G), (G, B), (W, G), (G, W) \\}\n", - "\\end{align*}\n", - "\n", - "c.\n", - "\n", - "$S = [0, \\frac{1}{3})$\n", - "\n", - "## Problem 3\n", - "\n", - "- $A \\cup B \\cup C = S$\n", - "- $P(A) = \\frac{3}{6}$\n", - "- $P(B) = \\frac{4}{6}$\n", - "- $P(A \\cup B) = \\frac{5}{6}$\n", - "\n", - "### a. Find $P(A \\cap B)$\n", - "\n", - "$P(A \\cap B) = P(A) + P(B) - P(A \\cup B) = \\frac{3}{6} + \\frac{4}{6} - \\frac{5}{6} = \\frac{2}{6}$\n", - "\n", - "### b. Do $A$, $B$ and $C$ form a partition of $S$?\n", - "\n", - "No. Proof by contridiction:\n", - "\n", - "Assume $A$, $B$ and $C$ form a partition of $S$. That would imply $P(S) = P(A) + P(B) + P(C) = 1$.\n", - "\n", - "$P(A) = \\frac{3}{6}$ and $P(B) = \\frac{4}{6}$, so $P(A) + P(B) = \\frac{7}{6}$\n", - "\n", - "$P(A) + P(B) > P(S)$ therefore contirdiction.\n", - "\n", - "$A$, $B$ and $C$ do not form a partition of $S$. $\\blacksquare$\n", - "\n", - "### c. Find $P(C \\setminus (A \\cup B))$\n", - "\n", - "\\begin{align*}\n", - "A \\cup B \\cup C = S \\\\\n", - "A \\cup B \\cup C \\setminus (A \\cup B) = S \\setminus (A \\cup B) \\\\\n", - "C \\setminus (A \\cup B) = S \\setminus (A \\cup B)\n", - "\\end{align*}\n", - "\n", - "Therefore $P(C \\setminus (A \\cup B)) = P(S \\setminus (A \\cup B))$.\n", - "\n", - "Notice $P(S \\setminus (A \\cup B)) = P((A \\cup B)^c)$, so $P(C \\setminus (A \\cup B)) = P((A \\cup B)^c)$\n", - "\n", - "So\n", - "\n", - "\\begin{align*}\n", - "P(C - (A \\cup B)) &= P((A \\cup B)^c) \\\\\n", - "&= P(S) - P(A \\cup B) \\\\\n", - "&= 1 - \\frac{5}{6} \\\\\n", - "&= \\frac{1}{6}\n", - "\\end{align*}\n", - "\n", - "### d. If $P(C \\cap (A \\cup B)) = \\frac{5}{12}, P(C) =$ ?\n", - "\n", - "$C = (C \\cap (A \\cup B)) \\cup C \\setminus (A \\cup B)$\n", - "\n", - "Therefore\n", - "\n", - "\\begin{align*}\n", - "P(C) &= P((C \\cap (A \\cup B)) \\cup C \\setminus (A \\cup B)) \\\\\n", - "&= P(C \\cap (A \\cup B)) + P(C \\setminus (A \\cup B)) \\\\\n", - "&= \\frac{5}{12} + \\frac{1}{6} \\\\\n", - "&= \\frac{7}{12}\n", - "\\end{align*}\n", - "\n", - "## Problem 4.\n", - "\n", - "### a. \n", - "Let $X$ be our sample space for the first row, and $Y$ the second roll. ($X \\cup Y = S$)\n", - "\n", - "$X = \\{ 1, 2, 3, 4, 5, 6 \\}$\n", - "\n", - "Let $x$ be the first dice row and $y$ the second dice roll. ($x \\in X$ and $y \\in Y$)\n", - "\n", - "The probability $x < y$ is $|X| - x$ or $6 - x$.\n", - "\n", - "So \n", - "\n", - "\\begin{align*}\n", - "\\sum_{x \\in X} (6 - x) &= \\sum_{x \\in X} 6 - \\sum_{x \\in X} x \\\\\n", - "&= 36 - (1 + 2 + 3 + 4 + 5 + 6) \\\\\n", - "&= 15\n", - "\\end{align*}\n", - "\n", - "Therefore $P(A) = \\frac{15}{36}$\n", - "\n", - "### b.\n", - "\n", - "Number of outcomes where 6 is only first: 1 * 5\n", - "Number of outcomes where 6 is only second: 1 * 5 \n", - "Number of outcomes where 6 is both: 1\n", - "\n", - "Therefore $P(B) = \\frac{11}{36}$\n", - "\n", - "## Problem 5." - ] - }, - { - "cell_type": "code", - "execution_count": 32, - "id": "e589ecc1", - "metadata": {}, - "outputs": [ - { - "data": { - "image/png": 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- "text/plain": [ - "
" - ] - }, - "metadata": { - "image/png": { - "height": 438, - "width": 696 - } - }, - "output_type": "display_data" - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "2.25091334378745\n" - ] - } - ], - "source": [ - "xmin = -4\n", - "xmax = 20\n", - "\n", - "x = np.linspace(xmin, xmax, 200)\n", - "y = np.exp(-1 * (x / 5))\n", - "\n", - "# ----------------------------------------------------\n", - "\n", - "plt.figure(figsize=(8, 5))\n", - "\n", - "# 1. Updated the LaTeX label to match your new math\n", - "plt.plot(x, y, color=\"crimson\", linewidth=2.5, label=r\"$y = e^{-x/5}$\")\n", - "\n", - "plt.xlim(xmin, xmax)\n", - "\n", - "# 2. Tightened the Y-limits so you can actually see the decay curve\n", - "plt.ylim(0, 2)\n", - "\n", - "plt.legend(fontsize=12, loc=\"upper right\") # Moved to upper right since curve decays downward\n", - "plt.show()\n", - "\n", - "print((1 - 2.71 ** (-3/5)) * 5)" - ] - }, - { - "cell_type": "markdown", - "id": "00afcbd1", - "metadata": {}, - "source": [ - "### a.\n", - "\n", - "Our sample space $S$ is the time at which the product breaks. The product could break immediatly ($T = 0$) or never break ($T \\to \\infty$)\n", - "\n", - "Therefore $S = [0, \\infty)$\n", - "\n", - "### b.\n", - "\n", - "$P(T \\geq 0) = e^{-\\frac{t}{5}} = 1$\n", - "\n", - "This makes sense, as this states that the probability the product doesnt break immediatly is 1. This would have to be the case, otherwise the product was broken to begin with, which is an impossible senario.\n", - "\n", - "$\\lim_{t \\to \\infty} P(T \\geq t) = \\lim_{t \\to \\infty} e^{-\\frac{t}{5}} = 0$ (skiping proof, can be easily seen in graph above)\n", - "\n", - "This means the as time goes on, the probability the product doesnt break approachs 0. This would imply the product will eventually break given infinite time.\n", - "\n", - "### c.\n", - "\n", - "$t_1 \\lt t_2 \\implies P(T \\geq t_1) \\geq P(T \\geq t_2)$.\n", - "\n", - "Proof\n", - "\n", - "\\begin{align*}\n", - "t_1 \\leq t_2 \\\\\n", - "-\\frac{t_1}{5} \\cdot -5 \\geq -\\frac{t_2}{5} \\cdot -5 \\\\\n", - "-\\frac{t_1}{5} \\geq -\\frac{t_2}{5} \\\\\n", - "e^{-\\frac{t_1}{5}} \\geq e^{-\\frac{t_2}{5}} \\\\\n", - "P(T \\geq t_1) \\geq P(T \\geq t_2) \n", - "\\end{align*}\n", - "\n", - "$\\blacksquare$\n", - "\n", - "This implies a more time means more probability the product may breakdown which makes intuitive sense.\n", - "\n", - "### d. Find the probability that the product breaks down within three years of the purchase time.\n", - "\n", - "\\begin{align*}\n", - "\\int_{0}^{3} e^{-\\frac{t}{5}} dt\\\\\n", - "[-5e^{-\\frac{t}{5}} + C]_0^3 \\\\\n", - "(-5e^{-\\frac{3}{5}} + C) - (-5e^{-\\frac{0}{5}} + C) \\\\\n", - "-5e^{-\\frac{3}{5}} + C + 5e^{-\\frac{0}{5}} - C \\\\\n", - "-5e^{-\\frac{3}{5}} + 5 \\\\\n", - "\\approx 2.251\n", - "\\end{align*}" - ] - }, - { - "cell_type": "markdown", - "id": "4609aba5-dfdc-4500-bec2-cfa2e544cca7", - "metadata": {}, - "source": [ - "### Example 1.17 - soln\n", - "\n", - "I roll a fair die twice and obtain two numbers X1 = result of the first roll and X2 = result\n", - "of the second roll. Given that I know X1 + X2 = 7, what is the probability that X1 = 4 or\n", - "X2 = 4?\n", - "\n", - "What is our sample space?\n", - "\n", - "\\begin{align*} \n", - "D = \\{1,2,3,4,5,6\\} \\\\\n", - "S = D \\times D\n", - "\\end{align*}\n", - "\n", - "\\begin{align*} \n", - "P(A|B) = \\frac{P(A \\cap B)}{P(B)}\n", - "\\end{align*}\n", - "\n", - "\\begin{align*} \n", - "A = \\{(x,y) \\mid x=4 \\lor y = 4 \\} \\\\\n", - "|A| = 12 \\\\\n", - "P(A) = \\frac{|A|}{|S|} \\\\\n", - "= \\frac{12}{36} = \\frac{1}{3}\n", - "\\end{align*}\n", - "\n", - "> NOTE my event set A may be overcounting \n", - "\n", - "\\begin{align*} \n", - "B = \\{(x,y) \\mid x+y = 7 , x,y \\in \\mathbb{N}\\} \\\\\n", - "|B| = 6 \\\\\n", - "P(B) = \\frac{1}{6} \\\\\n", - "\\end{align*}\n", - "\n", - "\\begin{align*} \n", - "P(A \\cap B) = \\frac{1}{18}\n", - "\\end{align*}\n", - "\n", - "\n", - "\\begin{align*} \n", - "B = \\{(x,y) \\mid x+y = 7 , x,y \\in \\mathbb{N}\\} \\\\\n", - "A = \\{(x,y) \\mid x=4 \\lor y = 4 \\} \\\\\n", - "P(A|B) = \\frac{P(A \\cap B)}{P(B)} \\\\\n", - "= \\frac{\\frac{1}{18} }{ \\frac{1}{6} } \\\\\n", - "= \\frac{6}{18} = \\frac{1}{3}\n", - "\\end{align*}\n" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "11163495-a6cc-43b9-bf41-02748b13d210", - "metadata": {}, - "outputs": [], - "source": [ - "\n" - ] - } - ], - "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/study/001_introduction-to-probability-statistics-and-random-processes/public/conditional_prob_tree.png new file 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