Probability Distributions #

Probability distributions are the bridge between: real-world randomness and mathematical modelling.

A random experiment produces outcomes. A random variable turns those outcomes into numbers. A probability distribution tells you how likely each number (or range of numbers) is.

Key takeaway: A distribution is a complete “story” about uncertainty: what values are possible, how likely they are, and how we summarise them (mean, variance).

flowchart TD
	PD["Probability<br/>distributions"] --> RV["Random<br/>variables"]
	PD["Probability<br/>distributions"] --> DS["Common<br/>distributions"]

	style PD fill:#90CAF9,stroke:#1E88E5,color:#000
	style RV fill:#90CAF9,stroke:#1E88E5,color:#000
	style DS fill:#90CAF9,stroke:#1E88E5,color:#000

AI/ML Connection #

Many ML models are probabilistic: they assume data (or errors) follow a distribution.
Loss functions often come from distribution assumptions: squared loss aligns with Gaussian noise.
Naïve Bayes (from the previous module) becomes practical once you can model: \( P(X\mid Y) \) using suitable distributions.

In practice: choosing a distribution is a modelling decision. It affects: prediction, uncertainty estimates, and what “rare” or “typical” means in your data.

Glossary #

Random variable: a rule that maps each outcome to a number.

pmf: for discrete variables, gives \( P(X=x) \) .

pdf: for continuous variables, probabilities come from areas: \( P(a\le X\le b)=\int_a^b f(x)\,dx \) .

cdf: accumulates probability up to x: \( F(x)=P(X\le x) \) .

References #

Devore: Probability and Statistics for Engineering and the Sciences (Ch. 3–4; plus t/chi-square/F later in the book)

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