Distributions

Compact study note.

Summary

Distribution means probability law induced by random variable. It can be described by PMF, PDF, CDF, or probability measure depending on variable type.^[1]

Prerequisites

• Random Variable

Notation and Assumptions

Notation: $P_X$ for induced distribution, $p_X$ for PMF, $f_X$ for PDF, and $F_X$ for CDF.

Essential Result

Choose the distribution by matching support, mechanism, independence assumptions, and parameter domains.

Small Example

Counts from fixed independent Bernoulli trials suggest a binomial model; waiting time to the first Poisson arrival suggests an exponential model.

Common Mistakes

• Choosing a distribution by curve shape alone.
• Mixing parameterizations without stating which one is used.

Connections

• Discrete Distributions
• Continuous Distributions
• Distribution Induced by a Random Variable

References