In Class Demos

Interactive notebooks for EE 503.

Running these

See Running Demos for how to launch in Colab or set up locally.

Week 1: Introduction to Probability

Randomness Demo — Can you fake randomness? Small-sample variability and statistical regularity.

Week 2: Conditioning and Independence

Conditioning Demo — Simpson’s paradox and Berkson’s bias: phantom correlations from aggregation and selection.
Binary Channel — Total probability, Bayes’ theorem, and channel capacity on the BSC.

Week 3: Counting and Combinatorics

Counting Demo — Birthday collisions and derangements converging to \(1/e\).

Week 4: Discrete Distributions

Discrete Explorer — Interactive PMF/CDF for the discrete BEG-CUP distributions.
Geometric Waiting Times — Memoryless property, counterexample, and the coupon collector.
Poisson Demo — Poisson as a binomial limit. The Poisson process and rare-event approximation quality.

Week 5: Continuous Densities and Bayesian Inference

Continuous Explorer — Interactive PDF/CDF for continuous BEG-CUP distributions.
Signal Detection — Hypothesis testing, ROC curves, \(d'\), and the \(\alpha\)/\(\beta\) tradeoff.
Thick Tails — Gaussian vs Cauchy: why the sample mean doesn’t converge.
Bayesian Conjugacy — An estimate without uncertainty is incomplete. Multi-armed bandits.

Week 7: Multiple Random Variables and Uncertainty

Joint Distributions — Discrete (trinomial) and continuous (bivariate Gaussian) joint PDFs.
Same Marginals — Marginals don’t determine the joint. What does: \(f_{XY} = f_X \cdot f_{Y|X}\).
Covariance — Covariance inflation of the sum. Diminishing returns of averaging correlated data.
Uncertainty — Uncertainty principles: can you measure when?

Week 8: Stochastic Convergence

Convergence Modes — What does it mean for a random sequence to converge?
Inequalities & WLLN — How moments limit tail probabilities. Probability bounds and WLLN.
Sample Statistics — How much can you trust a sample mean? Variance estimates and infinite variance.
Convergence in Distribution — CDF convergence pointwise: \(\text{Bin} \xrightarrow{d} \text{Pois}\), and a CLT preview.

Week 9: Conditional Expectation and Estimation

Conditional Expectation — \(E[Y \mid X = x]\) is a number. \(E[Y \mid X]\) is a random variable. JG conditionals and the mixture variance effect.
Total Theorems & Random Sums — Total expectation, total variance (EV + VE), and doubly random sums.
Optimal Estimators — MMSE and linear MMSE estimators. Conditional expectation as the best estimator.
EM Algorithm — Iterative maximum likelihood with hidden variables.

Week 10: Transformations and Monte Carlo

Density Transforms — How a PDF changes under \(Y = g(X)\), and the stretching factor \(|dx/dy|\).
CDF Sampling — Generating draws from any distribution via \(F^{-1}\).

Week 11: Central Limit Theorem and Confidence Intervals

Central Limit Theorem — Sampling and \(\bar{X}_n\). Standardized \(Z_n\) converges to standard normal \(N(0,1)\).
Convolution — Sum of independent random variables as convolution of densities.
Confidence Intervals — Random intervals: \(1-\alpha = P(\bar{X}_n - c \le \mu \le \bar{X}_n + c)\).