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For a continuous random variable X, a probability function f(x) is such that as δx → 0, P(x ≤ X ≤ x + δx ) = f(x) δx .
We have f(x) = F'(x) if F is a cumulative distribution function, and if the range of allowed values for X is [a, b] then

⌠ ⌡ b a  f(x)dx = 1 .

A function f_X for a given random variable X satisfying

P(X ∈ A) = ⌠ ⌡ A  fX(x) dx.

It is the limiting case of a relative frequency histogram as the sample size increases.