Probability and Statistics Course Overview

Course Logistics

• Attendance is mandatory for lectures and tutorials
• All course materials are available on Insendi, including slides, notes, problem sets, and lecture recordings
• The midterm exam contributes 30% to the final grade, and the final exam contributes 70%
• Problem sets provide practice problems similar to exam questions
• Tutorials review key concepts and problem set solutions; attendance is mandatory
• Office hours are held by the instructor and teaching assistant

Course Content

• This course provides an overview of the basics of probability and statistics
• Topics covered include:
  • Sets and events
  • Probability and counting
  • Probability distributions
  • Statistical inference

Resources

• Textbooks used in this course:
  • Introduction to Mathematical Statistics and Its Applications by Larsen and Marx (referred to as "LM" in lecture slides)
  • Introduction to Econometrics by Stock and Watson (referred to as "SW" in lecture slides)

Importance of Probability and Statistics

• Probability is the science of uncertainty
• Probability and statistics provide the foundation for any empirical study
• These skills are crucial for analyzing data and forming evidence-based conclusions

What is a Sequence?

• A sequence is a function that maps natural numbers to real numbers.
• We can write a sequence using the notation (an), (an)1, or (an)n=1, where an represents the value of the function at n.
• There are multiple ways to represent a sequence, for example using a formula, a recursive definition, or a graph.

What is Convergence?

• A sequence (an) converges to a real number 'a' if for any positive number 'ε', there exists a natural number 'N' where all terms after 'N' are within 'ε' distance from 'a'.
• This means that the terms of the sequence get arbitrarily close to 'a' as 'n' gets large enough.
• Another way to understand convergence is using ✏-neighborhoods: An ✏-neighborhood of 'a', denoted V✏(a), is an interval centered at 'a' with radius 'ε'.
• (an) converges to 'a' if for any ✏-neighborhood of 'a', there exists a point in the sequence after which all terms are in that neighborhood.
• The value of 'N' depends on the chosen 'ε', smaller neighborhoods generally require larger 'N'.

Proving Convergence

• To prove convergence, you need to show that for any 'ε' > 0, there exists an 'N' such that |an - a| < 'ε' for all n ≥ N.
• The Archimedean property helps with proofs: It states that the natural numbers are unbounded, meaning that for any positive number, there exists a natural number greater than that number.

Examples

• The sequence p1/n converges to 0.
• The sequence (n+1)/n converges to 1.

This quiz covers the fundamentals of probability and statistics as outlined in the course logistics. Key topics include sets, events, probability distributions, and statistical inference. Prepare for your midterm and final exams with comprehensive practice problems.