There are many interesting while at the same time very tricky problems in statistics. One famous question is that how many steps (expected) does it take to observe a given sequence (e.g. THTH, TTHH), if we flip a balanced coin?

This problem can be solved using (delay) renewal theory …

Generating functions is a very powerful way to find closed formula for sequences defined iteratively. I was so bored during the final week, so I went on internet for fun. Finally I found someone from Sydney University was asking for help on this question:

(a) If Ln=Ln-1+Ln-2 for …

Another interesting problem I met in statistic is: 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're allowed to choose first …

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 …