Welcome to STAT 425!

Probability theory (STAT 425), Fall 2023

Hi I’m Trevor

You can call me Trevor (if you like).

  • Pronouns: he/him/his

  • Hometown: Galesville, MD

  • Fun fact: I like to juggle (clubs, balls)

  • Office: Building 25 Room 236 (25-236)

  • Email: truiz01@calpoly.edu

  • Web: tdruiz.com

Warm-up

In groups of 3-4:

  1. go around and say your name, hometown, and current class standing
  2. write down 3 words, concepts, facts, statements, ideas, or phrases you associate with probability (can be one per person or by consensus, your choice)

After 5 minutes I’ll go around the room, ask you to share, and transcribe.

What is probability?

Based on our collective facts, can we…

  • identify common themes?

  • posit a rough definition or description consistent with our ideas?

Interpretations of probability

The expression \(Pr(\text{Millicent will go for a hike}) = 0.8\) represents the statement ‘there’s an 80% chance Millicent will go for a hike’. But what does that mean?

  • (subjective) we are 80% certain that Millicent will go for a hike

  • (objective) if we keep presenting Millicent with the same opportunity and circumstances, she’ll go for a hike on 8 out of every 10 occasions

Interpretations of probability

The subjective and objective interpretations of probability have names.

  • epistemic interpretation: probability statements indicate degrees of belief

  • aleatory interpretation: probability statements indicate frequencies of occurrence

The mathematics we’ll study are agnostic on the question of interpretation, though classical examples tend to align more naturally with the aleatory view (games of chance, dice rolls, etc.).

An informal definition

Probability is the mathematics of random events.

In this class we’ll develop a formal language for probability in terms of outcomes and events in a sample space:

  • outcome — result of an experiment or random process

  • event — statement about outcomes

  • sample space — collection of all possible outcomes

Using that language, we’ll construct familiar concepts of random variables, distributions, and their properties.

Since you already have some exposure/intuition from STAT 305 (or similar), the challenging part of this class for most of you will be the novel formalism and the level of problem-solving expected.

STAT 425 at a glance

Scope: we’ll cover probability axioms and properties, random variables, univariate and joint distributions and their properties.

Format: we’ll develop a set of course notes in class on the whiteboard following outlines that will be posted in advance of each class meeting; these will cover definitions, properties, theorems, and worked examples.

Materials: bring note-taking apparatus (pen/paper, tablet, etc.) and an internet-connected device to each class meeting. There is no required textbook, and the recommended text is available in StatLab (25-107B).

Assessments: weekly homeworks (7); monthly midterms (2); quarterly final exam (1).

Learning outcomes

(…with emphasis on key phrases)

  • use the axiomatic construction of probability to derive properties of probability measures and conditional probability measures, and apply definitions and properties to solve probability problems

  • construct probability models for discrete and continuous random variables, develop familiarity with common probability distributions, and use distribution functions to derive properties such as expectations and variances

  • determine joint distributions for collections of random variables, and use joint distribution functions to (a) derive properties such as covariance and correlation, (b) to determine conditional and marginal distributions, and (c) derive distributions of transformations and functions of one or more random variables

Note the frequent occurrence of the word use! I want you to understand and be able to use the concepts we discuss in class.

Assessment and evaluation

  • Homeworks (50%). Assigned/collected Thursdays; collaboration allowed; evaluated based on completeness, organization, and correctness of selected answers; lowest score dropped from final grade calculation.

  • Midterms (30%). Given in class Thursdays of weeks 4 and 8; note sheet allowed; evaluated based on completeness and correctness; higher score weighted more heavily in final grade calculation.

  • Final exam (20%). Administered as scheduled by Registrar; note sheet allowed; cumulative but emphasizes later material; evaluated based on completeness and correctness.

We aim to return graded work within one week, except for the final exam, which I’ll keep.

Letter grades (to within \(\pm 5\)): A: 90-100. B: 75-90. C: 60-75. D: 50-60.

Select policies

  • Time. Expect about 12-16 hours per week, including class time. Let me know if you’re exceeding this amount so I can help.
  • Collaboration. Allowed within the class only. Collaborations must reflect a legitimate shared effort, and names of collaborators should be written on submissions.
  • Attendance. Everyone is allowed two ‘free’ absences without notice at any time for any reason. Subsequent absences must be excusable and you should notify me by email.
  • Deadlines. Everyone is allowed two ‘free’ late homework submissions without notice at any time for any reason. Subsequent late submissions will be evaluated for 75% credit.

Select policies

  • Grades. Please report any discrepancies/errors in evaluation promptly (within 1 week of receiving an evaluated submission). Attempting to negotiate grades or seeking reevaluations after final grades are posted is not appropriate.
  • Communication. I prefer office hours and before/in/after class discussion for most matters. I try to respond to email within 48 weekday hours. Please wait a week before sending reminders, unless it’s time sensitive.
  • Conduct. Instances of academic dishonesty will be reported to OSRR and consequences may range from loss of credit to automatic failure, depending on the severity of the act.

General advice

I want each of you to succeed in this course and am here to help. I have a few recommendations:

  • spend 15-30 minutes skimming the suggested textbook sections each week in the StatLab

  • take advantage of the collaboration policy and work with classmates on homework

  • use your free absences and late submissions wisely

  • come to me early with any obstacles or difficulties you’re facing

  • take advantage of office hours; it’s time each week that I set aside specifically for you

For next time

  1. Complete intake survey
  2. Check the website before class for a lecture outline

© 2023 Trevor Ruiz