# 体育比赛中的一类概率分布问题A Probability Distribution Problem in Sports Games

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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.

1. 引言

$f\left(t\right)=\frac{1}{\sigma \sqrt{2\text{π}}}\mathrm{exp}\left[-\frac{{\left(t-\mu \right)}^{2}}{2{\sigma }^{2}}\right]$ (1)

$f\left(t\right)=\frac{K\left(x\right)}{\sigma \left(x\right)\sqrt{2\text{π}}}\mathrm{exp}\left[-\frac{{\left(t-x/v\right)}^{2}}{2{\sigma }^{2}\left(x\right)}\right]$ (2)

Figure 1. Histogram of flows for 18 km and 31 km (cited from [3] )

$t=\frac{x}{v}$ (3)

2. 比赛数据的统计分析

Figure 2. Diagrammatic sketch of the probability density function $f\left( t \right)$

Figure 3. Frequency histogram for x = 5 km water station

Figure 4. Frequency histogram for x = 10 km water station

Figure 5. Frequency histogram of the swimming results

3. 速度的概率密度函数

$f\left(v\right)=\frac{1}{\sigma \sqrt{2\text{π}}}\mathrm{exp}\left[-\frac{{\left(v-\mu \right)}^{2}}{2{\sigma }^{2}}\right]=N\left(\mu ,{\sigma }^{2}\right)$ (4)

$\mu =\stackrel{¯}{v}=\frac{{v}_{1}+{v}_{2}+,\cdots ,+{v}_{K}}{K}$ (5)

$\sigma =std\left(v\right)=\sqrt{\frac{{\left({v}_{1}-\stackrel{¯}{v}\right)}^{2}+{\left({v}_{2}-\stackrel{¯}{v}\right)}^{2}+,\cdots ,+{\left({v}_{K}-\stackrel{¯}{v}\right)}^{2}}{K-1}}$ (6)

4. 时间的概率密度函数

${f}_{Y}\left(y\right)=\left\{\begin{array}{cc}{f}_{X}\left[h\left(y\right)\right]|{h}^{\prime }\left(y\right)|,& y\in \left(a,b\right)\\ 0,\begin{array}{cccc}& & & \end{array}& 其他\end{array}$ (7)

$f\left(t\right)=\frac{x/\left(\mu t\right)}{\left(t\sigma /\mu \right)\sqrt{2\text{π}}}\mathrm{exp}\left[-\frac{{\left(t-x/\mu \right)}^{2}}{2{\left(t\sigma /\mu \right)}^{2}}\right]=x/\left(\mu t\right)\cdot N\left(x/\mu ,{\left(t\sigma /\mu \right)}^{2}\right)$ (8)

5. 结论

Figure 6. Comparison of normal probability density function and frequency histogram of speed

Figure 7. Comparison of normal-like probability density function and frequency histogram of time

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