Editing Unimodality (section)

===Other definitions===
Other definitions of unimodality in distribution functions also exist.

In continuous distributions, unimodality can be defined through the behavior of the [[cumulative distribution function]] (cdf).<ref name=Khinchin>{{cite journal|author=A.Ya. Khinchin|title=On unimodal distributions|journal=Trams. Res. Inst. Math. Mech.|publisher=University of Tomsk|volume=2|issue=2|year=1938|pages=1–7|language=ru}}</ref> If the cdf is [[convex function|convex]] for ''x''&nbsp;<&nbsp;''m'' and [[concave function|concave]] for ''x''&nbsp;>&nbsp;''m'', then the distribution is unimodal, ''m'' being the mode. Note that under this definition the [[uniform distribution (continuous)|uniform distribution]] is unimodal,<ref>{{Springer|title=Unimodal distribution|id=U/u095330|first=N.G.|last=Ushakov}}</ref> as well as any other distribution in which the maximum distribution is achieved for a range of values, e.g. trapezoidal distribution. Usually this definition allows for a discontinuity at the mode; usually in a continuous distribution the probability of any single value is zero, while this definition allows for a non-zero probability, or an "atom of probability", at the mode.

Criteria for unimodality can also be defined through the [[characteristic function (probability theory)|characteristic function]] of the distribution<ref name=Khinchin/> or through its [[Laplace–Stieltjes transform]].<ref>{{cite book|title=Random summation: limit theorems and applications|author=Vladimirovich Gnedenko and Victor Yu Korolev|isbn=0-8493-2875-6|publisher=CRC-Press|year=1996}} p.&nbsp;31</ref>

Another way to define a unimodal discrete distribution is by the occurrence of sign changes in the sequence of differences of the probabilities.<ref>{{cite journal|title=On the unimodality of discrete distributions |journal=Periodica Mathematica Hungarica|first=P. |last=Medgyessy|volume= 2| issue = 1–4 |pages=245–257|date=March 1972|url=http://www.akademiai.com/content/j5012306777g764n/ |doi=10.1007/bf02018665|s2cid=119817256 }}</ref> A discrete distribution with a [[probability mass function]], <math>\{p_n : n = \dots, -1, 0, 1, \dots\}</math>, is called unimodal if the sequence <math>\dots, p_{-2} - p_{-1}, p_{-1} - p_0, p_0 - p_1, p_1 - p_2, \dots</math> has exactly one sign change (when zeroes don't count).