#Probability

Week 1

Theme

This week saw the introduction of probability theory with the goal of setting out the axioms and approaching it with a more measure theoretic POV

Content covered

1. Finite Foundations
   1. We introduced the intuition behind probability and gave examples of experiments we’d like to investigate
   2. The Sample Space, Algebra, Finitely Additive Probability Measure were all defined
2. Probability Triples
   1. We then began to define more measure theoretic structures like σ-algebra, Probability Measure, as well as some properties like Probability Measure and Monotonicity of Probability Measure, Inclusion-Exclusion, Countable subadditivity.
   2. We then (as the subtitle suggests) defined what a Probability Space is.
3. Motivation and preliminaries
   1. Problem: Given $\Omega=[0,1]$ we want a σ-algebra $\mathcal{F}$ and Probability Measure $\mathbb{P}$ s.t.
      1. $I\in\mathcal{F}$ is an interval s.t. $I=|a,b|$ and;
      2. $\mathbb{P}((a,b))=b-a$ But because of Vitali, $\mathcal{F}=2^{\Omega}$ does not work, hence we need to construct some σ-algebra that works.
   2. To address this problem we then defined some additional structures like a semialgebra, along with a few propositions showing that intervals in $[0,1]$ are semialgebra’s
4. Extension Theorem
   1. After all the scaffolding was built, we stated The Extension Theorem

Week 2

Theme

We finished the proof of the extension theorem then introduced the uniform measure on $[0,1]$.

Content Covered

• Proved Extension Theorem
• Constructed the Uniform Random Variable

Week 3

Theme

Did some weird bullshit with the coin tossing space (I ain’t writing all that here) and defined Random Variables!

Content Covered

• Defined Coin Tossing Probability Space
• Defined Random Variable
  • Borel function
• Defined Independent rvs
• Looked at sequences of events:

  • First defined Increasing Events and Decreasing Events

  • Then we proved the Continuity of Probability

  • Covered the Infinitely often and proved an inequality.

    Week 4

    Theme

    This week was all about looking at sequences of events and their behaviour as we veer off to infinity!

Content

• Borel-Cantelli Lemma
• Kolmogorov 0-1 Law
• Defined Expectation
  • Defined some measure theory results
  • Lebesgue Integral
  • Integrable
  • Monotone Convergence Theorem
  • Fatou’s Lemma
  • Dominated Convergence Theorem
• Markov’s Inequality

Week 5

Theme

This week saw us continue with Expectation, specifically higher moments and Variance. Then we shifted gears to Convergence of rvs.

Content

• Expectation stuff
  • Defined Moment and Second Moment
  • Stated the Cauchy-Schwartz Inequality and proved it
    • Also showed that this inequality implies the following inclusion:$$\mathscr{L}^{p+1}\subseteq \mathscr{L}^{p}$$
  • We then defined Variance
  • Stated the Chebyshev Inequality and proved it.
  • Stated Jensen’s Inequality
• Convergence stuff
  • Pointwise Convergence
  • Almost Sure Convergence
  • In Probability Convergence
  • Pretty much proved most of this: in probability $\implies$ in distribution
  • Convergence in Expectation
• Integration for Independent rvs
  • Proved Independent, Independent

Week 6

Theme

This week was all about laws of large numbers. Then we capped it off with introducing distributions.

Content

• Stated and proved Weak Law of Large Numbers
• Defined what Identically Distributed and iid are.
• Stated and proved Beppo-Levi (Probability)
• Used this to then prove Strong Law of Large Numbers
• Defined Distribution
• Defined what a (Cumulative) Distribution Function is.
• Stated and proved equivalences b/w cdf and law.

Week 7

Theme

This week covered an array of topics that seeked to link the rv with the Distribution. We first covered the rv case then covered the Random Vector for existence results. Then we talked about how we can use Fubini-Tonelli to evaluate distributions of random vectors. We then finished it off with

Content

• Distribution discussion continued:
  • We showed in Distribution that any prob measure on $\mathbb{R}$ is the Distribution for some rv.
  • We then showed showed the Existence of Sequences of Independent rvs by:
    • first proving Existence of Sequences of Independent rvs.
    • Then proving the theorem itself.
• We then moved onto Fubini-Tonelli where we
  • Defined the Product σ-algebra
  • Showed the Product Measure exists.
  • Proved Fubini-Tonelli For Probability Measures.
• We capped off the discussion with:
  • Random Vectors
  • Showed how to find Distribution of Sum of Independent RVs using Convolutions.

Week 8

Theme

This week introduced Weak convergence/Convergence in Distribution. Wrapping up our conversation on convergence in probability. We then introduced tow more theorems that applied Law of Large Numbers.

Content

• Defined Weak convergence and Convergence in Distribution
  • Stated the Portmanteau’s Theorem
• Proved Convergence in Distribution
• Showed in probability $\implies$ in distribution
• Proved Skohorod’s Theorem