# Assuming the population is much larger than n it is

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Unformatted text preview: holds in the Markov inequality if and only if pY (0) + pY (c) = 1. Example 2.9.1 Suppose 200 balls are distributed among 100 buckets, in some particular but unknown way. For example, all 200 balls could be in the ﬁrst bucket, or there could be two balls in each bucket, or four balls in ﬁfty buckets, etc. What is the maximum number of buckets that could each have at least ﬁve balls? Solution: This question can be answered without probability theory, but it illustrates the essence of the Markov inequality. If asked to place 200 balls within 100 buckets to maximize the number of buckets with exactly ﬁve balls, you would naturally put ﬁve balls in the ﬁrst bucket, ﬁve in the second bucket, and so forth, until you ran out of balls. In that way, you’d have 40 buckets with ﬁve or more balls; that is the maximum possible; if there were 41 buckets with ﬁve or more balls then there would have to be at least 205 balls. This solution can also be approached using the Markov inequality as follows. Suppose t...
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## This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.

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