Unimodality

In mathematics, unimodality means possessing a unique mode. More generally, unimodality means there is only a single highest value, somehow defined, of some mathematical object. In statistics, a unimodal probability distribution or unimodal distribution is a probability distribution which has a single peak. The term "mode" in this context refers to any peak of the distribution, not just to the strict definition of mode which is usual in statistics. If there is a single mode, the distribution function is called "unimodal". If it has more modes it is "bimodal" (2), "trimodal" (3), etc., or in general, "multimodal". Figure 1 illustrates normal distributions, which are unimodal. Other examples of unimodal distributions include Cauchy distribution, Student's t-distribution, chi-squared distribution and exponential distribution. Among discrete distributions, the binomial distribution and Poisson distribution can be seen as unimodal, though for some parameters they can have two adjacent values with the same probability. Figure 2 and Figure 3 illustrate bimodal distributions. 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). If the cdf is convex for x < m and concave for x > m, then the distribution is unimodal, m being the mode. Note that under this definition the uniform distribution is unimodal, 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 of the distribution or through its Laplace–Stieltjes transform.