diff --git a/textbooks/reading/20260622012450_Introduction to Probability, Statistics, and Random Processes/ch1/flashcards/definitions.md b/textbooks/reading/20260622012450_Introduction to Probability, Statistics, and Random Processes/ch1/flashcards/definitions.md deleted file mode 100644 index fc76706..0000000 --- a/textbooks/reading/20260622012450_Introduction to Probability, Statistics, and Random Processes/ch1/flashcards/definitions.md +++ /dev/null @@ -1,222 +0,0 @@ -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/textbooks/reading/20260622012450_Introduction to Probability, Statistics, and Random Processes/ch1/flashcards/problems.md b/textbooks/reading/20260622012450_Introduction to Probability, Statistics, and Random Processes/ch1/flashcards/problems.md deleted file mode 100644 index e9ad6ee..0000000 --- a/textbooks/reading/20260622012450_Introduction to Probability, Statistics, and Random Processes/ch1/flashcards/problems.md +++ /dev/null @@ -1,17 +0,0 @@ -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