Computing Continuous Expectations

Probability · Axiom Academy

EXAMPLE Computing Continuous Expectations Calculate the expected value, second moment, and variance of a continuous random variable Excellent work! You've computed expectation and variance for a continuous random variable. Here's what we learned: Expected Value: For continuous X, E[X] = ∫ x·f(x) dx integrates over the entire domain, giving the center of mass of the distribution. Second Moment: E[X²] = ∫ x²·f(x) dx is needed to compute variance and represents the average squared value. Variance Formula: Var(X) = E[X²] - (E[X])² is the computational formula that's often easier than integrating (x - μ)²·f(x). Standard Deviation: σ = √Var(X) gives spread in the same units as X, making it more interpretable than variance. These calculations are fundamental to understanding the behavior of continuous distributions. Practice with different PDFs to build fluency!

This is the written version of the interactive lesson above. See the full Probability course.