From 26fa6b9723c1a6489773aef5e1c3d5f4bfa61093 Mon Sep 17 00:00:00 2001 From: Caleb Burke Date: Mon, 15 Jun 2026 00:31:45 -0700 Subject: [PATCH] adding flashcards! --- .../ch1/flashcards/definitions.md | 222 ++++++++++++++++++ .../ch1/flashcards/problems.md | 17 ++ .../ch2/end-of-chapter-problems.ipynb | 25 -- .../ch3/end-of-chapter-problems.ipynb | 25 -- .../ch4/end-of-chapter-problems.ipynb | 25 -- 5 files changed, 239 insertions(+), 75 deletions(-) create mode 100644 reading/000_Introduction to Probability, Statistics, and Random Processes/ch1/flashcards/definitions.md create mode 100644 reading/000_Introduction to Probability, Statistics, and Random Processes/ch1/flashcards/problems.md delete mode 100644 reading/000_Introduction to Probability, Statistics, and Random Processes/ch2/end-of-chapter-problems.ipynb delete mode 100644 reading/000_Introduction to Probability, Statistics, and Random Processes/ch3/end-of-chapter-problems.ipynb delete mode 100644 reading/000_Introduction to Probability, Statistics, and Random Processes/ch4/end-of-chapter-problems.ipynb diff --git a/reading/000_Introduction to Probability, Statistics, and Random Processes/ch1/flashcards/definitions.md b/reading/000_Introduction to Probability, Statistics, and Random Processes/ch1/flashcards/definitions.md new file mode 100644 index 0000000..fc76706 --- /dev/null +++ b/reading/000_Introduction to Probability, Statistics, and Random Processes/ch1/flashcards/definitions.md @@ -0,0 +1,222 @@ +State De Morgan's Law for $n$ sets +--- + +For any sets $A_1, A_2, \dots, A_n$: + +$$\left(\bigcup_{i=1}^n A_i\right)^c = \bigcap_{i=1}^n A_i^c$$ + +$$\left(\bigcap_{i=1}^n A_i\right)^c = \bigcup_{i=1}^n A_i^c$$ + +The complement of a union is the intersection of complements, and vice versa. + +=== + +State the Distributive Law for sets $A$, $B$, and $C$ +--- + +$$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$$ + +$$A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$$ + +=== + +State the inclusion-exclusion principle for a finite collection of sets $A_1, A_2, A_3, \dots A_n$ where $n=2$ + +--- + +$n = 2$ case: + +$|A \cup B| = |A| + |B| - |A \cap B|$ + +--- + +$n = 3$ case: + +$|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$ + +--- + +For a finite collection of sets $A_1, A_2, A_3, \dots A_n$, we have + +$\left| \bigcup_{i=1}^n A_i \right| = \sum_{i=1}^n |A_i| - \sum_{i < j} |A_i \cap A_j| + \sum_{i < j < k} |A_i \cap A_j \cap A_k| - \dots + (-1)^{n-1} |A_1 \cap A_2 \cap \dots \cap A_n|$ + +=== + +For the function + +$$f: A \to B$$ + +State the following of $f$: + +1. Domain +2. Co-domain +3. Range + +--- + +1. $A$ +2. $B$ +3. Set of all possible outputs of $f$ (not necessarily $B$) + +=== + +State definition for each the following: + +1. Random experiment +2. Outcome +3. Sample space +4. Event + +--- + +1. A **random experiment** is a process by which we observe something uncertain +2. An **outcome** is a result of a random experiment +3. The **sample space** $S$ is the set of all possible outcomes +4. An **event** is a subset of the sample space + +=== + +State the Axioms of Probability + +--- +**Axioms of Probability** + +1. For any event $A$, $P(A) \geq 0$ +2. $P(S) = 1$ +3. If $A_1, A_2, A_3, \dots$ are disjoint events, then $P(A_1 \cup A_2 \cup A_3 \cup \dots) = P(A_1) + P(A_2) + P(A_3) + \dots$ + +=== + +In a finite sample space $S$, where all outcomes are equally likely, what is the probability of any event $A$? + +--- + +$P(A) = \frac{|A|}{|S|}$ + +=== + +What is $P(A^c)$ in terms of $P(A)$? +--- + +$$P(A^c) = 1 - P(A)$$ + +Follows from the axioms: $A$ and $A^c$ are disjoint, and $A \cup A^c = S$, so $P(A) + P(A^c) = P(S) = 1$. + +=== + +Define conditional probability $P(A|B)$ +--- + +The probability of $A$ given $B$ (when $P(B) > 0$): + +$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$ + +In the equally-likely case: + +$$P(A|B) = \frac{|A \cap B|}{|B|}$$ + +=== + +State the chain rule of probability for $n$ events +--- + +$$P(A_1 \cap A_2 \cap \cdots \cap A_n) = P(A_1)\,P(A_2|A_1)\,P(A_3|A_1,A_2)\cdots P(A_n|A_1,A_2,\dots,A_{n-1})$$ + +Each factor conditions on all previously listed events. + +=== + +What conditions must hold for three events $A$, $B$, $C$ to be independent? +--- + +All four conditions must hold: + +1. $P(A \cap B) = P(A)P(B)$ +2. $P(A \cap C) = P(A)P(C)$ +3. $P(B \cap C) = P(B)P(C)$ +4. $P(A \cap B \cap C) = P(A)P(B)P(C)$ + +Pairwise independence alone is **not** sufficient. + +=== + +What does it mean for $n$ events $A_1, A_2, \dots, A_n$ to be independent? +--- + +Every subset of the events must satisfy the product rule. That is, for all $i < j < k < \dots$: + +$$P(A_i \cap A_j) = P(A_i)P(A_j)$$ +$$P(A_i \cap A_j \cap A_k) = P(A_i)P(A_j)P(A_k)$$ +$$\vdots$$ +$$P(A_1 \cap A_2 \cap \cdots \cap A_n) = \prod_{i=1}^n P(A_i)$$ + +=== + +If $A_1, A_2, \dots, A_n$ are independent, what is $P(A_1 \cup A_2 \cup \cdots \cup A_n)$? +--- + +$$P(A_1 \cup A_2 \cup \cdots \cup A_n) = 1 - \prod_{i=1}^n (1 - P(A_i))$$ + +Derived by taking the complement (none of the events occur) and using independence. + +=== + +What is the difference between disjoint and independent? + +--- + +- Disjoint: $A$ and $B$ cannot occur at the same time. $A \cap B = \empty$ +- Independent: $A$ does not give any information about $B$ + +=== + +State the Law of Total Probability using event $B$ and its complement +--- + +$$P(A) = P(A|B)\,P(B) + P(A|B^c)\,P(B^c)$$ + +=== + +State the general Law of Total Probability for a partition of the sample space +--- + +If $B_1, B_2, B_3, \dots$ is a partition of $S$, then for any event $A$: + +$$P(A) = \sum_i P(A \cap B_i) = \sum_i P(A|B_i)\,P(B_i)$$ + +=== + +State Bayes' Rule for two events $A$ and $B$ +--- + +For $P(A) \neq 0$: + +$$P(B|A) = \frac{P(A|B)\,P(B)}{P(A)}$$ + +=== + +State Bayes' Rule when $B_1, B_2, \dots$ form a partition of $S$ +--- + +For any event $A$ with $P(A) \neq 0$: + +$$P(B_j|A) = \frac{P(A|B_j)\,P(B_j)}{\displaystyle\sum_i P(A|B_i)\,P(B_i)}$$ + +The denominator expands $P(A)$ via the Law of Total Probability. + +=== + +State the meaning of conditionally independent between events $A$ and $B$ given event $C$. Note $P(C) \gt 0$. + +--- + +$$P(A \cap B \mid C) = P(A \mid C)P(B \mid C)$$ + +=== + +State the multiplication rule for $P(A \cap B)$ +--- + +$$P(A \cap B) = P(A|B)\,P(B) = P(B|A)\,P(A)$$ + +Rearrangement of the definition of conditional probability. diff --git a/reading/000_Introduction to Probability, Statistics, and Random Processes/ch1/flashcards/problems.md b/reading/000_Introduction to Probability, Statistics, and Random Processes/ch1/flashcards/problems.md new file mode 100644 index 0000000..e9ad6ee --- /dev/null +++ b/reading/000_Introduction to Probability, Statistics, and Random Processes/ch1/flashcards/problems.md @@ -0,0 +1,17 @@ +For any event $A$, prove $P(A^c) = 1 - P(A)$ + +--- + +$$ +1 = P(S) += P(A \cup A^c) += P(A) + P(A^c) +$$ + +=== + +Prove $P(A \setminus B) = P(A) - P(A \cap B)$ + +--- + +TODO diff --git a/reading/000_Introduction to Probability, Statistics, and Random Processes/ch2/end-of-chapter-problems.ipynb b/reading/000_Introduction to Probability, Statistics, and Random Processes/ch2/end-of-chapter-problems.ipynb deleted file mode 100644 index 5ffd73b..0000000 --- a/reading/000_Introduction to Probability, Statistics, and Random Processes/ch2/end-of-chapter-problems.ipynb +++ /dev/null @@ -1,25 +0,0 @@ -{ - "cells": [ - { - "cell_type": "code", - "execution_count": null, - "id": "43cc5069", - "metadata": {}, - "outputs": [], - "source": [] - }, - { - "cell_type": "markdown", - "id": "d706c044", - "metadata": {}, - "source": [] - } - ], - "metadata": { - "language_info": { - "name": "python" - } - }, - "nbformat": 4, - "nbformat_minor": 5 -} diff --git a/reading/000_Introduction to Probability, Statistics, and Random Processes/ch3/end-of-chapter-problems.ipynb b/reading/000_Introduction to Probability, Statistics, and Random Processes/ch3/end-of-chapter-problems.ipynb deleted file mode 100644 index 5ffd73b..0000000 --- a/reading/000_Introduction to Probability, Statistics, and Random Processes/ch3/end-of-chapter-problems.ipynb +++ /dev/null @@ -1,25 +0,0 @@ -{ - "cells": [ - { - "cell_type": "code", - "execution_count": null, - "id": "43cc5069", - "metadata": {}, - "outputs": [], - "source": [] - }, - { - "cell_type": "markdown", - "id": "d706c044", - "metadata": {}, - "source": [] - } - ], - "metadata": { - "language_info": { - "name": "python" - } - }, - "nbformat": 4, - "nbformat_minor": 5 -} diff --git a/reading/000_Introduction to Probability, Statistics, and Random Processes/ch4/end-of-chapter-problems.ipynb b/reading/000_Introduction to Probability, Statistics, and Random Processes/ch4/end-of-chapter-problems.ipynb deleted file mode 100644 index 5ffd73b..0000000 --- a/reading/000_Introduction to Probability, Statistics, and Random Processes/ch4/end-of-chapter-problems.ipynb +++ /dev/null @@ -1,25 +0,0 @@ -{ - "cells": [ - { - "cell_type": "code", - "execution_count": null, - "id": "43cc5069", - "metadata": {}, - "outputs": [], - "source": [] - }, - { - "cell_type": "markdown", - "id": "d706c044", - "metadata": {}, - "source": [] - } - ], - "metadata": { - "language_info": { - "name": "python" - } - }, - "nbformat": 4, - "nbformat_minor": 5 -}