who have at least $100$ and $2000$ valid player-selected odds above $m_\text{min}$. According to the results in Fig. \[fig:wager\_distr\_top\], after removing the inequality the wager distributions can still be approximated by log-normal distributions, but some deviation can be observed. Similarly, the odds distributions again follow truncated shifted power-law distributions after removing the inequality. These results demonstrate that the shape of the distributions we obtained in the above sections is not a result of the inequality of the number of bets.

![\[fig:wager\_distr\_top\] The wagers obtained from random sampling of top gamblers’ bets still present log-normal distributions, although there are some observable deviations. ](figure5.pdf){width="\columnwidth"}

Now our question becomes whether the conclusion regarding the distribution at the population level can be extended to the individual level. Here due to the limitation of data, we will only discuss the wager distribution. Analyzing the individual distribution of top gamblers, we find that although heavy-tailed properties can be widely observed at the individual level, only a small proportion of top gamblers presents log-normal distributed wagers. Other distributions encountered include log-normal distributions, power-law distributions, power-law distributions with exponential cutoff, pair-wise power-law distributions, irregular heavy-tailed distributions, as well as distributions that only have a few values. The diversity 