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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.
- Spring '08
- The Land