Random Variables

ABSTRACT

This directory covers the mathematical framework for mapping experimental outcomes to numerical values. It transitions from simple counting to functional analysis through probability distributions and specific statistical models.

Knowledge Map

Foundations

• Random Variables: The formal definition of $X:S\to \mathbb{R}$ and how to calculate the expectation $E[X]$ using outcomes or distribution values.
• Distributions: The study of how probability mass is allocated across a sample space and the application of the Law of Total Probability.
• Uniform vs Non-Uniform Distributions: Distinguishing between “fair” systems (equally likely) and “biased” or “weighted” systems.

Specific Discrete Models

• Binomial Distribution: Models the number of successes in a fixed number of $n$ independent Bernoulli trials.
  • Key formula: $P(k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$
• Geometric Distribution: Models the number of trials required to reach the first success in an infinite sequence.
  • Key formula: $P(k)=(1-p{)}^{k-1}p$

Quick Reference: Which Model to Use?

Core Identities

• Sum of a Distribution: ${\sum }_{a\in S}p(a)=1$
• Expectation of $X$: $E[X]=\sum x\cdot P(X=x)$
• Binomial Expectation: $E[X]=np$
• Geometric Expectation: $E[X]=1/p$

Related Toolkits

• Laws of Probability: Provides the underlying logic for independence and conditional probability used in these distributions.
• Expected Value and Analysis: Advanced techniques like Linearity of Expectation and Case Analysis used to solve complex problems involving these variables.