We form the Jacobi theta distribution through discrete integration of exponential random variables over an infinite inverse square law surface. It is continuous, supported on the positive reals, has a single positive parameter, is unimodal, positively skewed, and leptokurtic. Its cumulative distribution and density functions are expressed in terms of the Jacobi theta function. We describe asymptotic and log-normal approximations, inference, and a few applications of such distributions to modeling.

翻译：我们通过在无限反向平方法律表面的指数随机变量的离散集成形成雅各比的分布。它是连续的,在正正正正方形上得到支持,有一个单一正参数,是单式的,正偏斜的,和列普托库尔特的。它的累积分布和密度函数以雅各比特塔函数表示。我们描述的是微量和对数正常近似值、推论,以及这种分布在建模方面的几个应用。