Probability density function

Consider a probability model $(𝜉, F, ℙ)$ . The probability density function $𝑓 𝑋$ of a general random variable $𝑋 \to 𝜉$ of a measure space $(X, Σ, 𝜇)$ is the Radon-Nikodym derivative

𝑓 𝑋 = 𝑑 𝑋 * ℙ 𝑑 𝜇

where $𝑋 * ℙ$ is the Pushforward measure.

Real random variable

For the simple case of a real random variable the probability density function is simply the derivative of the cumulative distribution function.

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