Unimodal distribution

A unimodal distribution, under the density or mass-function convention, has probabilities that rise toward one central region and then fall away. For an absolutely continuous probability distribution, this means it has a probability density function that is nondecreasing up to a mode and nonincreasing afterward. This is the single-maximum meaning of unimodality; the chosen density is a one-dimensional quasiconcave function. A plateau of maximizing points is allowed.

For a discrete original example on outcomes0,1,2and3, choose probabilities0.1,0.4,0.4and0.1. The probability mass function rises, has two adjacent maximizing values, then falls. It is unimodal under the weak monotonicity definition despite not having a unique mode. In contrast, separated high-probability regions with a genuine dip between them are not one mode plateau.

Unimodality does not imply symmetry or equality of arithmetic mean, median and mode. For the exponential distribution with density $p(x)=\lambda e^{-\lambda x}$ on $x\ge 0$, $\lambda >0$, the mode is0, the mean is $1/\lambda$ and the median is $\log (2)/\lambda$. The density decreases from its boundary maximum. The NIST handbook gives the family and these summary characteristics, using a scale parameter reciprocal to this rate parameter.^[ref]

Nor is unimodality generally preserved by a mixture distribution. Equal mixing of uniform distributions on $[0,1]$ and $[3,4]$ gives density1/2 on those two intervals and zero in between. Each component has a flat unimodal density, while the mixture has separated peaks.

A density is defined only up to changes on sets of measure zero. Thus the distributional statement concerns existence of a representative with the required shape, rather than artificial isolated spikes added to one displayed formula. In higher dimensions, several notions of unimodality are used; one convex-level-set formulation requires convex superlevel sets, as in the quasiconcave function definition.^[ref] The shape convention should be specified before extending a one-dimensional conclusion to another setting.