Probability Distributions

February 22, 2026

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.

Random Variables

February 22, 2026

Random Variables #

A random variable is a way to attach numbers to outcomes of a random experiment.

It lets us move from: “what happened?” to: “what number should we analyse?”

Key takeaway: A random variable is a function from the sample space to real numbers. Once you define the random variable clearly, the rest (pmf/pdf/cdf, mean, variance) becomes systematic.

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

RV --> T["Types"]
T --> RV1["Discrete<br/>RVs"]
T --> RV2["Continuous<br/>RVs"]

RV --> F["PMF / PDF / CDF"]
RV --> S["Mean / Variance<br/>Covariance"]
RV --> J["Joint & Marginal<br/>distributions"]
RV --> X["Transformations"]

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

style T fill:#CE93D8,stroke:#8E24AA,color:#000
style F fill:#CE93D8,stroke:#8E24AA,color:#000
style S fill:#CE93D8,stroke:#8E24AA,color:#000
style J fill:#CE93D8,stroke:#8E24AA,color:#000
style X fill:#CE93D8,stroke:#8E24AA,color:#000
style RV1 fill:#CE93D8,stroke:#8E24AA,color:#000
style RV2 fill:#CE93D8,stroke:#8E24AA,color:#000

1) Definition #

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

Common Probability Distributions

February 22, 2026

Common Probability Distributions #

Once you can describe a random variable using a pmf or pdf, the next step is to use named distributions that appear repeatedly in real data and in ML models.

Key takeaway: Named distributions give you ready-made probability models for common patterns: binary outcomes, counts, and measurement noise.

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

DS --> DIS["Discrete"]
DS --> CON["Continuous"]

DIS --> D1["Bernoulli"]
DIS --> D2["Binomial"]
DIS --> D3["Poisson"]

CON --> D4["Normal<br/>(Gaussian)"]
CON --> D5["t / Chi-square / F<br/>(intro)"]

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

style DIS fill:#CE93D8,stroke:#8E24AA,color:#000
style CON fill:#CE93D8,stroke:#8E24AA,color:#000

style D1 fill:#C8E6C9,stroke:#2E7D32,color:#000
style D2 fill:#C8E6C9,stroke:#2E7D32,color:#000
style D3 fill:#C8E6C9,stroke:#2E7D32,color:#000
style D4 fill:#C8E6C9,stroke:#2E7D32,color:#000
style D5 fill:#C8E6C9,stroke:#2E7D32,color:#000

1) Bernoulli distribution (binary) #

Use when: one trial has two outcomes (success/failure).