Unimodal function

Unimodal function

In mathematics, a function "f"("x") between two ordered sets is unimodal if for some value "m" (the mode), it is monotonically increasing for "x" ≤ "m" and monotonically decreasing for "x" ≥ "m". In that case, the maximum value of "f"("x") is "f"("m") and there are no other local maxima.

Examples of unimodal function:

*Quadratic polynomial
*Logistic map
*Tent map

Function f(x) is "S-unimodal" if its Schwartzian derivative is negative for all x e 0.

In probability and statistics, a "unimodal probability distribution" is a probability distribution whose probability density function is a unimodal function, or more generally, whose cumulative distribution function is convex up to "m" and concave thereafter (this allows for the possibility of a non-zero probability for "x"="m"). For a unimodal probability distribution of a continuous random variable, the Vysochanskii-Petunin inequality provides a refinement of the Chebyshev inequality. Compare multimodal distribution.

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