PUMaC 2016 · 团队赛 · 第 8 题
PUMaC 2016 — Team Round — Problem 8
题目详情
- (6) Alice has 100 balls and 10 buckets. She takes each ball and puts it in a bucket that she chooses at random. After she is done, let b be the number of balls in the i th bucket, for i 10 ∑ 2 1 ≤ i ≤ 10. Compute the expected value of b . i i =1
解析
- Observe that for each ball, the expected number of balls in its bucket is 1 + 99 · = . The 10 10 1 ∑ 2 expected value of 0 b is just the expected value of the sum over all balls of the number i i =1 of balls in their bucket. By linearity of expectation, this is equal to 100 times the expected 109 number of balls in a ball’s bucket. Thus, the answer is 100 · = 1090 . 10 Problem written by Eric Neyman. √