Random Variables

TL;DR

A random variable is a variable that can take on different values based on the outcome of a random event or experiment. The chance of an outcome happening from this specific experiment is determined by a probability distribution function. Random variables are used to model and analyze uncertain or unpredictable phenomena, and they are a fundamental concept in probability theory and statistics.

Types of Random Variables

• Discrete Random Variables
  • Bernoulli Random Variable
  • Binomial Random Variable
• Continuous Random Variables
  • Uniform Random Variable
  • Normal (Gaussian) Random Variable
  • Exponential Random Variable

Properties of Random Variables

• Expected Value (Mean)
  • Discrete Expected Value
  • Continuous Expected Value
• Variance and Standard Devision
  • Discrete Variance
  • Continuous Variance
• Moment Generating Function
  • Discrete MGF
  • Continuous MGF

Joint Random Variables

• Joint Probability Distribution
  • Discrete Joint Distribution
  • Continuous Joint Distribution
• Covariance and Correlation
  • Discrete Covariance
  • Continuous Covariance

Applications of Random Variables

• Decision Making under Uncertainty
  • Risk Analysis
  • Reliability Engineering
• Simulation and Modeling
  • Monte Carlo Simulation
  • Queuing Theory Models
• Statistical Inference