Probability Density Functions

Probability · Axiom Academy

LESSON Probability Density Functions Understanding how probability is distributed over continuous values A function f(x) is a valid probability density function if it satisfies two key properties: Non-negativity: f(x) must be greater than or equal to zero for all x. Total area equals 1: The integral of f(x) over all possible values must equal 1, ensuring total probability is 1. To find the probability that X falls in an interval [a, b], we integrate the PDF over that interval: The probability is represented by the area under the curve f(x) between a and b. This is fundamentally different from discrete random variables, where we sum probabilities. A crucial distinction: the PDF f(x) itself is not a probability, but a probability density . The value f(x) can actually be greater than 1, as long as the total area integrates to 1. Think of density as "probability per unit length" - just like physical density (mass per unit volume).

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