Init

.github/workflows/publish.yml

@@ -0,0 +1,25 @@
+on:
+  workflow_dispatch:
+  push:
+    branches: main
+
+name: Quarto Publish
+
+jobs:
+  build-deploy:
+    runs-on: ubuntu-latest
+    permissions:
+      contents: write
+    steps:
+      - name: Check out repository
+        uses: actions/checkout@v4
+
+      - name: Set up Quarto
+        uses: quarto-dev/quarto-actions/setup@v2
+
+      - name: Render and Publish
+        uses: quarto-dev/quarto-actions/publish@v2
+        with:
+          target: gh-pages
+        env:
+          GITHUB_TOKEN: ${{ secrets.GITHUB_TOKEN }}

Foundation of Probability Theory.md

@@ -163,12 +163,17 @@ is defined as a mapping that satisfies the following three conditions:
 
 ::: {.callout-important}
 
-A probability function tell how the probability of occurrence is distributed over the set of events $\mathbb{B}$.In this sense we speak of a distribution of probability.
+- A probability function tell how the probability of occurrence is distributed over the set of events $\mathbb{B}$.In this sense we speak of a distribution of probability.
+
+![](https://img12.360buyimg.com/ddimg/jfs/t1/232346/23/5558/25512/656c2680F5d41484e/4bbc3248c4c4ad99.jpg){width=50% fig-align="center"}
+
+- For a given measurable space (𝑆, 𝔹), many different probability functions can be defined. The goal of econometrics and statistics is to find a probability function that most accurately describes the underlying DGP. This probability function is usually called the true probability function or true probability distribution model.
 
 :::
 
 
 
+
 ## Methods of Counting
 ## Conditional Probability
 ## Bayes' Theorem

_book/Foundation of Probability Theory.html

@@ -329,6 +329,12 @@ <h2 data-number="2.4" class="anchored" data-anchor-id="fundamental-probability-l
 </div>
 <div class="callout-body-container callout-body">
 <p>A probability function tell how the probability of occurrence is distributed over the set of events <span class="math inline">\(\mathbb{B}\)</span>.In this sense we speak of a distribution of probability.</p>
+<div class="quarto-figure quarto-figure-center">
+<figure class="figure">
+<p><img src="https://img12.360buyimg.com/ddimg/jfs/t1/232346/23/5558/25512/656c2680F5d41484e/4bbc3248c4c4ad99.jpg" class="img-fluid figure-img" style="width:50.0%"></p>
+</figure>
+</div>
+<p>For a given measurable space (𝑆, 𝔹), many different probability functions can be defined. The goal of econometrics and statistics is to find a probability function that most accurately describes the underlying DGP. This probability function is usually called the true probability function or true probability distribution model.</p>
 </div>
 </div>
 </section>

_book/search.json

@@ -74,7 +74,7 @@
     "href": "Foundation of Probability Theory.html#fundamental-probability-laws",
     "title": "2  Foundation of Probability Theory",
     "section": "2.4 Fundamental Probability Laws",
-    "text": "2.4 Fundamental Probability Laws\n✅ To assign a probability to an event \\(A \\in S\\), we shall introduce a probability function, which is a function or a mapping from an event to a real number (0,1).\n✅ To assign probabilities to events, complements of events,unions and intersections of events, we want our collection of events to include all these combinations of events.\n✅Such a collection of events is called an \\(\\sigma\\)-field(\\(\\sigma\\) algaebra) of subsets of the sample space S,which constitude the domain of the probability function.\n\n\n\nimage.png\n\n\n[💬 Definition 10. Sigma Algebra] A \\(sigma(\\sigma) algebra\\),denoted by \\(\\mathbb{B}\\) , is a collection of subsets(events) of S that satisfies the following three conditions:\n\n\\(\\emptyset \\in \\mathbb{B}\\)i.e., the empty set is in \\(\\mathbb{B}\\).\nIf \\(A \\in \\mathbb{B}\\), then \\(A^{c} \\in \\mathbb{B}\\).(i.e., \\(\\mathbb{B}\\) is closed under complements)\nIf \\(A_1,A_2, \\ldots \\in \\mathbb{B}\\), then \\(\\bigcup_{i=1}^{\\infty} A_i \\in \\mathbb{B}\\).(i.e., \\(\\mathbb{B}\\) is closed under countable unions)\n\n\n\n\n\n\n\nImportant\n\n\n\n\n\\(\\sigma\\)-algebra is a collection of events in \\(S\\)(subset) that satisfies certain properties and constitutes the domain of a probability function.\nA \\(\\sigma\\) -field is a collection of subsets in \\(S\\) , but itself is not a subset of \\(S\\) . In contrast, the sample space \\(S\\) is only an element of a \\(\\sigma\\) -field.\nThe pair \\((S,\\mathbb{B})\\) is called a measurable space.So for a specific sample space \\(S\\), we can have different \\(\\sigma\\)-algebra \\(\\mathbb{B}\\).\n\n\n\n[💬 Definition 11. Probability Function] Suppose a random experiment has a sample space \\(S\\) and an associated \\(\\sigma\\) -field \\(\\mathbb{B}\\) . A probability function \\[\nP:\\mathbb{B} \\to [0,1]\n\\]\nis defined as a mapping that satisfies the following three conditions:\n\n\\(0 \\le P(A) \\le 1 for all A \\in \\mathbb{B}\\) .\n\\(P(S) = 1\\).\nIf \\(A_1,A_2, \\ldots \\in \\mathbb{B}\\) are mutually exclusive, then \\(P\\left( \\bigcup_{i=1}^{\\infty} A_i \\right) = \\sum_{i=1}^{\\infty} P(A_i)\\).🚩The probability of the union of mutually exclusive events is the sum of the probabilities of the individual events.\n\n\n\n\n\n\n\nImportant\n\n\n\nA probability function tell how the probability of occurrence is distributed over the set of events \\(\\mathbb{B}\\).In this sense we speak of a distribution of probability."
+    "text": "2.4 Fundamental Probability Laws\n✅ To assign a probability to an event \\(A \\in S\\), we shall introduce a probability function, which is a function or a mapping from an event to a real number (0,1).\n✅ To assign probabilities to events, complements of events,unions and intersections of events, we want our collection of events to include all these combinations of events.\n✅Such a collection of events is called an \\(\\sigma\\)-field(\\(\\sigma\\) algaebra) of subsets of the sample space S,which constitude the domain of the probability function.\n\n\n\nimage.png\n\n\n[💬 Definition 10. Sigma Algebra] A \\(sigma(\\sigma) algebra\\),denoted by \\(\\mathbb{B}\\) , is a collection of subsets(events) of S that satisfies the following three conditions:\n\n\\(\\emptyset \\in \\mathbb{B}\\)i.e., the empty set is in \\(\\mathbb{B}\\).\nIf \\(A \\in \\mathbb{B}\\), then \\(A^{c} \\in \\mathbb{B}\\).(i.e., \\(\\mathbb{B}\\) is closed under complements)\nIf \\(A_1,A_2, \\ldots \\in \\mathbb{B}\\), then \\(\\bigcup_{i=1}^{\\infty} A_i \\in \\mathbb{B}\\).(i.e., \\(\\mathbb{B}\\) is closed under countable unions)\n\n\n\n\n\n\n\nImportant\n\n\n\n\n\\(\\sigma\\)-algebra is a collection of events in \\(S\\)(subset) that satisfies certain properties and constitutes the domain of a probability function.\nA \\(\\sigma\\) -field is a collection of subsets in \\(S\\) , but itself is not a subset of \\(S\\) . In contrast, the sample space \\(S\\) is only an element of a \\(\\sigma\\) -field.\nThe pair \\((S,\\mathbb{B})\\) is called a measurable space.So for a specific sample space \\(S\\), we can have different \\(\\sigma\\)-algebra \\(\\mathbb{B}\\).\n\n\n\n[💬 Definition 11. Probability Function] Suppose a random experiment has a sample space \\(S\\) and an associated \\(\\sigma\\) -field \\(\\mathbb{B}\\) . A probability function \\[\nP:\\mathbb{B} \\to [0,1]\n\\]\nis defined as a mapping that satisfies the following three conditions:\n\n\\(0 \\le P(A) \\le 1 for all A \\in \\mathbb{B}\\) .\n\\(P(S) = 1\\).\nIf \\(A_1,A_2, \\ldots \\in \\mathbb{B}\\) are mutually exclusive, then \\(P\\left( \\bigcup_{i=1}^{\\infty} A_i \\right) = \\sum_{i=1}^{\\infty} P(A_i)\\).🚩The probability of the union of mutually exclusive events is the sum of the probabilities of the individual events.\n\n\n\n\n\n\n\nImportant\n\n\n\nA probability function tell how the probability of occurrence is distributed over the set of events \\(\\mathbb{B}\\).In this sense we speak of a distribution of probability.\n\n\n\n\n\nFor a given measurable space (𝑆, 𝔹), many different probability functions can be defined. The goal of econometrics and statistics is to find a probability function that most accurately describes the underlying DGP. This probability function is usually called the true probability function or true probability distribution model."
   },
   {
     "objectID": "Foundation of Probability Theory.html#methods-of-counting",

_quarto.yml

@@ -20,5 +20,5 @@ format:
   pdf:
     documentclass: scrreprt
 
-
-
+execute:
+  freeze: auto