Math 466 - Theory of Statistics - Fall 2023
ChatGPT-4 Chats related to the course
This class uses ChatGPT in instruction (in mostly non-essential, education enhancing ways). The following is a light-minded commentary on the tool.
It's well
established now that the way you put a question often
determines not only the answer you'll get, but the type of
answer possible. So ... a mechanical answerer, geared to
produce the ultimate revelations in reference to anything
you want to know, might have unsuspected limitations.
From
Ask A Foolish Question
- by Robert Sheckley
- Sample space, random variables for coin tosses of a biased coin
- Distribution of the sum of two dice
- Distribution of the sum of $n$ dice in "closed" form
- Three levels of expected value theory
- Sum of Bernoulli and normal
- Does pairwise independence imply independence? - very interesting, ChatGPT recovers by itself from initial mistakes, and proves a theorem
- Summary of probability generating functions (pgf)
- Basic t-test
- Summary of moment generating functions and outline of the proof of the Central Limit Theorem - includes commentary on the gaps if we do not use characteristic functions, and highlights the role of Levy" Theorem
- Paired t-test
- Decomposing sample variance, sample size 4, explicitly found iid of standard normals, one of which is the sample mean - We inflicted pain on each other (Chat GPT and I) but the final product has a touch of genius!
- An simple example leading to F-distribution
- Discussion of sample mean as a Minimum Variance Unbiased Estimator (MVUE)
- Derivation of MLE estimator of Geometric Distribution - this is an extended example, with CLT, Delta Method and more
- Solving inequalities with parameters - uses the 'Wolfram' plugin, version of 9-27-2023
- Delta Method
- Three estimators of uniform distribution, efficiency analysis - uses Wolfram plugin
- Comparison of terms like variance, sample variance, sample standard deviation, standard error - these are often confused, and notations vary across books and papers, so it should be useful
- A proof of Cramer-Rao inequality and comments on the score function
- An attempt at calculating Cramer-Rao Lower Bound (CRLB) for uniform distributions - this is a story of failure, but, on the other hand, I do not know a clear path to a solution.
- The basics of conditional expectation
- Given a normal vector (X,Y), what is E[Y|X=x]?
AI-related movie recommendations, trailers, related topics
- The Artifice Girl
- An interview with an AI robot -
claim made that
- Chat GPT v.4 is 10 times more intelligent than Chat GPT v. 3.5
- that it has IQ of about 155 (Einstein IQ astimated at 160)
- in 3 years AI will be thousands times more intelligent than an average human
- A darker view of AI
Miscellaneus Topics
Music
- Music, generation of sheet music - uses a plugin, capability of 9-27-2023
- Music theory, analysis of 'Country roads' by John Denver, a transcription by Per-Olov Kindgren of Pachelbel's "Canon in D" for solo guitar.
Mathematica (Wolfram) plugin - capability of 9-27-2023
Biology - uses plugins, capability of 9-27-2023
- Krebs cycle --- drawing diagram and generating MATLAB code
Interpretation of images
This feature as of 10-05-2023, requires a cellphone and the ChatGPT app installed.- A conversation about a photo I took on campus (a screenshot, click to enlarge)
Plugin development
Game playing
This uses the 'Advanced Data Analysis' flavor of ChatGPT.- Tic-tac-toe - first attempt, constructs min-max move generator in Python, unable to run Python but code runs locally
- Tic-tac-toe - second attempt, generates Python code to train a neural player; could not get it to run locally yet, issues apparently not related to ChatGPT