Interactive Statistics Platform
Master Probability Visually.
Understand distributions, limit theorems, and sampling statistics through an interactive curriculum, real-time sandboxes, and visual quizzes.
Guided Curriculum
4 chapters taking you from probability foundations to advanced continuous distributions and limit theorems.
14 Distributions
Interactive mathematical widgets for normal, uniform, geometric, gamma, beta, log-normal, Student's T, and more.
Quiz Engine
Test your knowledge with multiple choice, calculation problems, and parameter curve-matching challenges.
Sampling Simulator
Run real-time animated sampling distributions to prove the Law of Large Numbers and Central Limit Theorem.
Foundations of Probability
A Random Variable is a variable whose values are outcomes of a random process. They are categorized as either discrete or continuous. To describe how probabilities are distributed, we use function mappings:
- Probability Density Function (PDF) f(x) for continuous variables, describing the density of probability at any point (area under the curve represents probability).
- Probability Mass Function (PMF) P(X = x) for discrete variables, giving the exact probability of specific outcomes.
- Cumulative Distribution Function (CDF) F(x) = P(X ≤ x), representing the accumulated probability up to value x.
The core bridge between PDF and CDF is calculus: the PDF is the derivative of the CDF, i.e., f(x) = dF(x)/dx.
Expected Value and Variance
The Expected Value (Mean, μ) represents the long-term average outcome of a random variable. The Variance (σ²) measures the spread or dispersion of the distribution around its expected value.
Discrete Probability Models
Discrete distributions apply to scenarios with countable outcomes. Here are the 5 core models:
- Bernoulli: A single trial with success probability p.
- Binomial: Count of successes in n independent Bernoulli trials.
- Poisson: Number of events in a fixed time/space interval under constant rate λ.
- Geometric: Number of failures before the first success.
- Hypergeometric: Successes in draws without replacement from a finite population.
Continuous Probability Models
Continuous models describe variables that can take any value in an interval. Standard distributions include:
- Uniform: Flat probability across [a, b].
- Normal: Symmetric bell curve representing central averages.
- Exponential: Models memoryless waiting times between events.
- Gamma & Beta: Highly flexible shape-parameter families. Beta is defined on [0, 1].
- Chi-Square: Distribution of sum of squared normal variables.
- Log-Normal: Skewed variable whose log is normally distributed.
- Student's T: Symmetric bell curve with fat tails for small sample sizes.
- Weibull: Widely used for failure rates and material lifetime analysis.
Sampling & Limit Theorems
Limit theorems form the bedrock of statistical inference, explaining what happens when we draw many samples:
- Law of Large Numbers (LLN): As the number of trials increases, the sample average converges to the theoretical expected value (μ).
- Central Limit Theorem (CLT): Regardless of the shape of the underlying distribution, the distribution of the sample mean converges to a Normal Distribution as the sample size (n) becomes large (typically n ≥ 30).
What
When
How
Probability Calculator
1 — Model
2 — Operation
3 — Result
Double Distribution Comparison
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This quiz consists of 5 questions including conceptual choices, formula calculations, and interactive curve parameter-matching puzzles. Complete the quiz with 100% accuracy to log it in your progress tracker.
Convergence Statistics (CLT Proof)
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