Probability density function ($f_X(x)$)

A probability density function is a nonnegative measurable function $f_X$ satisfyingIts integral over the whole sample space is one.

A normalizing constant makes a nonnegative integrable function into a probability density function: if $0<Z=\int h<\infty$, then $h/Z$ is a probability density function.

A density ratio compares two probability density functions relative to the same underlying measure. More generally it is a Radon-Nikodym derivative when one probability measure has absolute continuity of measures with respect to the other.

A probability density is log-concave when its logarithm is a concave function on its support. Tangent lines to a differentiable concave log density lie above its graph, giving exponential envelopes for rejection sampling.

For a probability density function $f$ on $\mathbb{R}$ with Lipschitz bound $L>0$, the triangular lower envelope $f(x+u)\ge (f(x)-L|u|{)}_{+}$ gives $f(x{)}^2\le L$. If $f$ is strictly positive everywhere, the inequality is strict. Equality forces the entire probability density function to coincide with the triangular envelope, which vanishes outside a finite interval.

A joint probability density $f_{X,Y}$ satisfiesIts coordinate integrals are the marginal densities, and integrating it over a rectangle gives the corresponding joint distribution function.

If $Y=g(X)$ for a differentiable bijection, thenThis is the change of variables formula applied to probability density.