Continuous Distributions Summary

Probability · Axiom Academy

SUMMARY Continuous Distributions Let's review the key continuous probability distributions and their properties. Uniform Distribution: Models equally likely outcomes over an interval . All values have the same probability density. Exponential Distribution: Models waiting times and time between events. Has the memoryless property. Normal Distribution: The bell curve. Describes many natural phenomena and is central to the CLT. Gamma Distribution: Generalizes the exponential distribution. Models the sum of independent exponential random variables. Beta Distribution: Defined on . Perfect for modeling proportions, probabilities, and percentages. Statistical Testing Distributions Chi-Squared: Sum of squared standard normals. Used in goodness-of-fit tests and variance estimation. Student's t: Used when sample size is small and population variance is unknown. Has heavier tails than normal. F-Distribution: Ratio of two chi-squared distributions. Used for comparing variances and ANOVA. Key Relationships Between Distributions Gamma to Exponential: When shape parameter , Gamma becomes Exponential Gamma to Chi-Squared: is equivalent to Normal to Chi-Squared: Sum of squared standard normals gives Chi-Squared to t: If and , then Chi-Squared to F: If and , then Sum of Normals: Sum of independent normals is also normal with combined mean and variance Choosing the Right Distribution Bounded interval with equal probability: Use Uniform distribution

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