Dating back to the time when I was a sophomore, I read a very interesting problem from "Waming", which is a magazine of math department of USTC (University of Science and Technology of China).

One day Guigu Zi (a famous ancient Chinese sophist, the teacher of Sun Bin and Pang …

I have to deal with Modified Bessel functions of the first kind frequently in my research. Here I list some useful properties of them for future reference.

Suppose we flip a coin which has probability 0.7 to be head again and again and two people choose two different sequences of length 3 (e.g., THH). The people whose sequence appears first wins. If you are allowed to choose first, which sequence will you choose?

I discussed …

Suppose you toss a symmetric dice. You are allowed to quit the game and get money which equals the total points you get at any time if 6 has never showed up. Whenever 6 shows up, the game is over and you get nothing. For example, if the first three …