From eb14b81c3cfa3af4a46b2b1a53a6d4ff255be2bb Mon Sep 17 00:00:00 2001 From: eggy Date: Thu, 23 Mar 2023 18:20:23 -0400 Subject: [PATCH] ece108: add probability intro --- docs/1b/ece108.md | 38 ++++++++++++++++++++++++++++++++++++++ 1 file changed, 38 insertions(+) diff --git a/docs/1b/ece108.md b/docs/1b/ece108.md index e0c2283..d2e2542 100644 --- a/docs/1b/ece108.md +++ b/docs/1b/ece108.md @@ -846,3 +846,41 @@ $$|A\cup B\cup C|=|A| + |B| + |C| - (|A\cap B| + |A\cap C| + |B\cap C|)-|A\cap B If $B$ is a subset of $A$, the dimension of $B$ is related to that of $A$. $$B\subseteq A\implies|B|=|A|-|\overline B|$$ + +## Probability + +!!! definition + - An **experiment** is an event that has a number of outcomes. + - **Elementary events** are the outcomes of an experiment compose the set of all events. + - An **event** $E$ is a subset of the sample space $S$, which is the **certain event**. + - The **null event** is the empty set. + - Sets of events are **mutually exclusive** if they are disjoint. + - Elementary events are **equiprobable** if they are equally probable. + - A **uniform probability distribution** on $S$ is such that all elementary events are equiprobable. + +A **probability distribution function (PDF)** $Pr$ converts the elements of the powerset of all outcomes to a real number — its probability. + +$$Pr:\mathcal P(S)\to\mathbb R,0\leq P(A)\leq 1$$ + +A PDF must have, if $S$ is the sample space: + +- $\forall A\subseteq S,Pr\{A\}\geq 0$ +- $Pr\{S\}=1$ +- The union of all mutually exclusive sets is the sample space + +A **discrete probability distribution** is such that the sample space is a countable set. + +For all $A\subseteq S$, the probability of event $A$ is the sum of the probabilities of all elementary events in $A$. + +- $Pr\{A\}=\sum_{e\in A}Pr\{\{e\}\}$ +- $Pr\{\empty\}=0$ +- $Pr\{A'\}=1-Pr\{A\}$ + +Adding events together can never decrease their probability, and the sum of all probabilities must equal $1$ such that $\text{rang}(Pr)\subseteq[0,1]$. + +$$A\subseteq B\subseteq S\implies Pr\{A\}\leq Pr\{B\}$$ + +The **inclusion-exclusion principle** also applies. + +$$Pr\{A\cup B\}=Pr\{A\}+Pr\{B\}-Pr\{A\cup B\}$$ +