Probability Density Function - Formal Definition

Formal Definition

(This definition may be extended to any probability distribution using the measure-theoretic definition of probability.)

A random variable X with values in a measure space (usually Rn with the Borel sets as measurable subsets) has as probability distribution the measure X_∗P on : the density of X with respect to a reference measure μ on is the Radon–Nikodym derivative:

That is, f is any measurable function with the property that:

for any measurable set .