体育比赛中的一类概率分布问题

A Probability Distribution Problem in Sports Games

在有关体育比赛的数学建模问题中,运动员的跑步、游泳等比赛时间通常被假设为服从正态分布。本文通过对若干马拉松及铁人三项比赛成绩进行统计和理论分析后发现:运动员的平均速度服从一般的正态分布,但是运动员到达赛程中某一固定点(比如终点)所用的时间并不服从正态分布,而是服从一个变参数的、右偏的类正态分布。文中推导了时间概率分布密度函数的数学表达式,该结论可应用于类似比赛或其它应用领域的相应数学建模问题。

In mathematical modelling problems relating to sports games, the running or swimming time of athletes is usually assumed to follow a normal distribution. This paper statistically and theoreti-cally analyzes the data collected from some real Marathon and Triathlon games, and shows that in a sports event the athletes’ average speed has a general normal distribution, whereas the time taken by the athletes arriving at a fixed point (e.g., the destination) in the race course does not follow such type of distribution, but rather a parameter-varying right-skewed normal-like distribution. Moreover, the mathematical formula for the probability density function of the athletes’ race time is derived, and the results obtained may also be applied to the corresponding mathematical modelling problems of similar sports games or other application areas.