EXPLORE DISTRIBUTIONS

📚 BASICS

What is a Probability Distribution?

A probability distribution describes how likely different outcomes are. Think of it as a map showing where probability "mass" is concentrated.

• Random Variable (X): A quantity that can take different values randomly
• PDF/PMF: Probability at each point (density vs mass)
• CDF: Cumulative probability up to a point
• Mean (μ): Expected value / center of mass
• Variance (σ²): Spread / how scattered the values are

📊 DISCRETE vs CONTINUOUS

Two Types of Distributions

Discrete: Countable outcomes (0, 1, 2, 3...)

• Bernoulli: Single coin flip
• Binomial: Multiple coin flips
• Poisson: Count of rare events

Continuous: Infinite possible values

• Normal: Bell curve for natural phenomena
• Uniform: Equal probability everywhere
• Exponential: Time between events

🎯 REAL-WORLD EXAMPLES

Where You See These

Normal: Heights, test scores, measurement errors

Binomial: Quality control (n defects), free throw success

Poisson: Website visits per hour, calls to a help desk

Exponential: Time until next earthquake, product lifetime

Uniform: Random number generators, lottery draws

Bernoulli: Single yes/no event, pass/fail test

🧮 KEY FORMULAS

The Math You Need

Expected Value (Mean): E[X] = Σ(x × P(x))

Variance: Var(X) = E[(X - μ)²]

Standard Deviation: σ = √Var(X)

68-95-99.7 Rule (Normal):

• 68% within 1σ of mean
• 95% within 2σ of mean
• 99.7% within 3σ of mean