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## What is the Tine Distribution?

The little known **Tine distribution** (sometimes called symmetric triangular distribution) made an entry in the Index to the Distributions of Mathematical Statistics [1] as:

The Annals of Mathematical Statistics [2] contains an expanded definition:

Rinne [3] defines Tine’s distribution as the distribution of two independent and identically distributed 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*).

He notes it is also called Simpson’s distribution, after Thomas Simpson (1710-1761) who is thought to be the first to suggest the distribution.

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 (μt) and the standard deviation ((sigma) t) of a person distribution, and the average value (μo) and standard deviation (σo) of a

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

## References

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

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

[3] 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