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一道coin flip题目
[版面:金融工程][首篇作者:roccopass] , 2019年08月27日11:35:57 ,829次阅读,0次回复
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发信人: roccopass (roccopass), 信区: Quant
标  题: 一道coin flip题目
发信站: BBS 未名空间站 (Tue Aug 27 11:35:57 2019, 美东)

Expected number of coin flips until all cars move to end of array?

Imagine that we have an array of length 2n, where the first n entries are a
C (representing a toy car) and the remaining n entries are empty.
Additionally, we have n fair coins labeled 1 through n, where coin i
corresponds to car C_i in the array.

On each timestep, we flip all n coins. If coin i comes up as heads, then car
C_i moves forward in the array by one spot, but only if it is not blocked
by another car directly in the slot in front of it. Else, if blocked or the
coin comes up tails, car C_i does nothing.

The question has two parts:
1.What is the expected number of timesteps until the n-th car reaches the
end of the array (reaches slot 2n)?
2.What is the expected number of timesteps until all of the n cars have
moved from the first n slots of the array to the last n slots?

第1问比较显然,n-th car移动一位的期望投掷次数是2,移动n位那就是2n;
第2问感觉好复杂,因为其它car能否移动取决于右边car的移动情况,也就是说这个
bernoulli试验中的概率p是一直在变化的,由其它bernoulli试验的结果所影响。比方
说(n-1)-th第一次投掷能够往右移动一位的概率是1/4,第二次投掷能够往右移动的概
率是3/4*1/4+1/4*1/2=5/16,第三次移动的概率就比较棘手了。不知道大家有没有好的
思路,any idea appreciated!
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