###### Abstract
The ITU recommendation uses the measured rainfall rate exceeding 0.01% of the time and then extrapolates attenuation at other time percentages. Therefore, the ITU method provides identical results for two sites whose rate values were exceeded 0.01% of the time, even if, at other percentages of time, they differed widely. ITU-R618-13 uses specific attenuation for the calculation, as given by the ITU-R Rec. 838-3 [17]. This has been found to show good agreement with measured data at frequencies below 100 GHz. At higher frequencies, drop size distribution (DSD) becomes a significant factor, and physically based models using Mie scattering and the DSD models exhibit better fit to measured data [7; 20], though experimental work at these frequencies is scarce [7]. DSD models include Marshall-Palmer [21], JOSS [22], and Weibull [23]. Our method employs the specific attenuation as given by the ITU-R Rec. 838-3 [17], however, in contrast, we use the probability distribution function for rain rate in order to develop a mathematical expression for symbol error rate. This term is a function of rain rate, mean and variance, specific attenuation, as well as the symbol SNR in the absence of rain. An ITU recommendation also includes a dynamic approach, but it differs from ours. ITU-R Rec. P.1853-2 [19] provides methods for synthesizing time series of tropospheric impairments, including rain attenuation on earth-space paths as well as single terrestrial paths. There, attenuation is derived from the assumption that the long-term statistics for attenuation are the lognormal distribution and recommends using ITU-R P.618-13 [18] to obtain that. In contrast, our method relies on the assumption that the rain rate, rather than attenuation, is lognormally distributed, and the attenuation is obtained as a function of rate. ITU-R Rec. P.1853-2 involves a synthesis of a white Gaussian noise series and is a numerical procedure, while our method provides a closed mathematical term for symbol error rate. ITU-R.Rec.837-7 [24] provides a method for predicting rainfall rate statistics with a one-minute integration time. This can be used to provide the rate and the mean in our method, since our method does not predict a specific rate, but rather the rate appears as a parameter in the final expression. Our work is based on a derivation of the probability distribution function [25], which also provided data for the mean and variance; thus, for purposes of this paper, those were used. We focus on outdoor long-distance communication as relevant to satellite-ground links. The challenge is that the symbol error probability (SEP) is a non-linear function in terms of signal to noise ratio and attenuation, which is correlated with the rain rate. Therefore, employment of the present statistical methods for rain rate and attenuation in order to calculate SEP is less precise, especially for the high frequencies that will be used in the near future. In this work, we calculate the correct average symbol error probability. This could be a crucial improvement because, when using the standard method, the values of the symbol error rate (SER) under-estimate the performance in comparison to reality. With 6G technology and, especially, with extensive use of IoT, a precise calculation of error will be of overriding importance. The paper is organized as follows. In the first section, some background is presented regarding rain and the statistics or rain rate. Next, the probability density function for rain is given. The introductory background concludes with details of rain attenuation. In the following section, we find a closed mathematical expression for the average symbol error as a function of the probability density function of rain. A numerical comparison of methods is presented, and the paper ends with a discussion of the results.