= Unimodal distribution
A \b[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 $x\geq0$, $\lambda>0$, 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.<Exponential Distribution in the NIST e-Handbook>[§1.3.6.6.7, Probability Density Function and summary statistics]{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.<Convex Optimization by Stephen Boyd and Lieven Vandenberghe>[§3.4.1, superlevel characterization]{ref} The shape convention should be specified before extending a one-dimensional conclusion to another setting.
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