Probability Density Function

A probability density function describes a continuous random variable through probability per unit length, not probability at a point.

Definition

For a continuous random variable X, the probability density function (PDF) f(x) satisfies f(x) ≥ 0 and ∫ f(x) dx = 1 over the whole line. Probabilities are areas under the curve: P(a ≤ X ≤ b) = ∫ from a to b of f(x) dx.

Why a single point has probability zero

The area under a curve over an interval of zero width is zero, so P(X = c) = 0 for any exact value c. This is why, for continuous variables, ≤ and < give the same probability. The density is not the probability of a value; it is a local rate.

Relationship to the CDF

The density is the derivative of the cumulative distribution function, f(x) = dF/dx, and conversely F(x) = ∫ from −∞ to x of f(t) dt. Where the CDF rises steeply the density is high; where the CDF is flat the density is near zero.

Expectation from a density

Means and variances become integrals: E[X] = ∫ x f(x) dx and Var(X) = ∫ (x − E[X])² f(x) dx. The same integral pattern computes the expectation of any function g(X) as ∫ g(x) f(x) dx.

Change of variables

If Y = g(X) with g monotonic, the density transforms with a Jacobian: f_Y(y) = f_X(x) / |g'(x)|. Forgetting the Jacobian is a common error when transforming simulated samples, and it silently distorts every downstream statistic.