Statistics provides the mathematical framework for reasoning under uncertainty. These notes cover the core toolkit: how data behaves, how to test claims, how to model relationships, and how to update beliefs as evidence arrives. Start with a cluster overview and follow links into specifics.
Probability & Foundations
- Random Variable - the formal bridge from outcomes to numbers; entry point for all distribution and inference concepts
- Conditional Probability - updating probability with new information by restricting the sample space
- Bayes’ Rule - reversing conditional probability to compute the probability of a cause given an effect
- Expected Value - the probability-weighted average outcome of a random variable
- Variance and Covariance - measuring spread and co-movement between variables
Probability Distributions
- Probability Distributions - overview of discrete and continuous families; PMFs, PDFs, and CDFs
- Binomial Distribution - counting successes in fixed independent trials
- Geometric Distribution - trials until the first success
- Poisson Distribution - event counts in a fixed interval at constant rate
- Exponential Distribution - waiting time between Poisson events
- Normal Distribution - the Gaussian bell curve; foundation for inference
Statistical Inference
- Central Limit Theorem - why sample averages converge to normality
- Law of Large Numbers - why the sample mean converges at all, and how that differs from the CLT
- Sampling and Sampling Distributions - the distribution of a statistic, and standard error
- Hypothesis Testing - null and alternative hypotheses, p-values, significance levels, Type I/II errors, and power
- Confidence Intervals - interval estimation, and what the coverage claim does not say
- The t-Distribution and t-Tests - heavier tails, degrees of freedom, and the one-sample, two-sample, and paired tests
- Maximum Likelihood Estimation - choosing parameters that make observed data most probable
- Bootstrap and Resampling - estimating a sampling distribution by resampling the data you already have
Regression & Modeling
- Regression Fundamentals - multiple linear regression, OLS estimation, residual analysis, and R-squared
- Simple Linear Regression - fitting a straight line by minimizing squared errors
Bayesian Methods
- Bayesian Inference - prior, likelihood, posterior; updating beliefs with observed data via Bayes’ theorem
The full file listing follows below, generated automatically by Quartz.