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## What is the Tine Distribution?

The little known **Tine distribution** , sometimes called the *symmetric triangular distribution, is a continuous probability distribution shaped like a triangle. *It is also called *Simpson’s distribution*, after Thomas Simpson (1710-1761) who is thought to be the first to suggest the distribution [1]. The distribution was not mentioned in the literature again until R. Schmidt’s 1934 article in the Annals of Mathematical Statistics [2]. Schmidt was the first to call it the “tine distribution” (a tine is a slender projecting point).

The tine distribution made an entry in the Index to the 1958 *Distributions of Mathematical Statistics *[3] as:

Rinne [3] defines the Tine distribution as the distribution of two independent and identically distributed (i.i.d.) uniform variables (i.e., the convolution of two uniform distributions):

*X*_{1}, *X*_{2} iid∼ *UN*(*a, b*) ⇒ *X* = *X*_{1} + *X*_{2} ∼ *TS*(2 *a* + *b*, *b*).

A *convolution *is an operation on two functions (*f* and *g*) that produces a third function (), which expresses how the shape of one function is modified by the other.

The distribution isn’t widely known. In fact, if you try and Google “tine Distribution” you’ll be redirected (at the time of writing) to pages on “time distribution” instead. It’s also not often used (most likely because it isn’t well known!), but there are a few specific use cases. For example, this Google patent for an “Authentication device and authentication method” includes the tine distribution as a threshold measure.

The threshold value determination part 22 is Mahalanobis prescribed | regulated by the Mahalanobis distance prescribed | regulated by the mean value and the standard deviation of a person distribution, and the average value and standard deviation of a

Google patent for an authentication devicetine distribution. To match the distance, the threshold value Xth is determined.

Note that the patent makes reference to a “person distribution” which is most likely the author’s name for the distribution of biometric authentication data.

References

[1] Rinne, H. Location–Scale Distributions Linear Estimation and Probability Plotting Using MATLAB. Online: http://geb.uni-giessen.de/geb/volltexte/2010/7607/pdf/RinneHorst_LocationScale_2010.pdf

[2] Haight, F. (1958). Index to the Distributions of Mathematical Statistics. National Bureau of Standards Report.

[3] Schmidt, R. Statistical Analysis of One-Dimensional Distributions. Annals of Mathematical Statistics. 5:33. 1934