Distributions

Probability distributions describe how random variables behave, assigning likelihoods to different outcomes. From simple binary events to complex continuous patterns, distributions are essential tools for modeling data and uncertainty. In this section, we explore key families of distributions that form the backbone of statistical inference and machine learning. 📊

Learning Objectives

Learning objectives of the Distributions section.

• Bernoulli
• Gaussian
• Beta

Summary Table

Summary of the Distributions section.

Distribution	Type	Support	Parameters	PDF / PMF	Common Use Case
Bernoulli	Discrete	\(x \in \{0, 1\}\)	\(p\) (success probability)	\(p_X(x) = \begin{cases} p & \text{if } x = 1, \\ 1 - p & \text{if } x = 0 \end{cases}\)	Binary outcomes (e.g., success/failure, yes/no)
Gaussian (Normal)	Continuous	\(x \in (-\infty, \infty)\)	\(\mu\) (mean), \(\sigma^2\) (variance)	\(f_X(x)=\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}\)	Modeling natural phenomena, basis of CLT
Beta	Continuous	\(x \in (0, 1)\)	\(\alpha\), \(\beta\)	\(f_X(x) = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha - 1}(1 - x)^{\beta - 1}\)	Bayesian priors for probabilities, modeling proportions