WassupAI

The LLN guarantees that as you collect more data, the sample average converges to the true population average. It provides the mathematical proof for why casinos always win in the long run and justifies using large datasets to estimate unknown parameters.

Independence extends beyond events to variables. If knowing X tells you nothing about Y, they are independent. We define this formally using joint densities, allowing us to simplify complex models by treating components separately.

How do variables move together? Covariance measures the direction of the relationship, while Correlation standardizes it between -1 and 1. We will learn to quantify the strength and direction of linear associations between random variables.

From a joint distribution, we extract specific views. Marginals isolate one variable by "summing out" the others, while Conditionals update the probabilities of one variable given that another has taken a specific value.

Real-world problems involve multiple interacting variables. Joint distributions quantify the probability of two or more variables occurring simultaneously, creating the mathematical foundation for analyzing relationships and complex systems.

The Uniform distribution models "maximum uncertainty," where all intervals are equally likely. The Beta distribution is a flexible shape-shifter often used in Bayesian statistics to model uncertainty about probabilities or proportions themselves.

The Exponential distribution models memoryless waiting times, like radioactive decay. The Gamma distribution generalizes this to the time required for multiple events to occur, making both essential for reliability engineering and queuing theory.

The "Bell Curve" is the cornerstone of statistics. Defined by mean and variance, it appears naturally in error analysis and nature. We’ll master the Z-score and the 68-95-99.7 rule to estimate probabilities quickly and accurately.

For continuous variables, probability is the area under a curve, not a point. The PDF describes the relative likelihood of values, where the total area equals 1. It is the tool for modeling continuous measurements like time, distance, or temperature.

We explore the standard models of discrete probability: Bernoulli (single trials), Binomial (success counts), Geometric (waiting times), and Poisson (rare events). These templates allow us to model standard patterns found in science and business.