25 in dening the condence interval it makes the


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Unformatted text preview: he balls are distributed among the buckets in some particular way. The probability experiment is to select one of the buckets at random, with all buckets having equal probability. Let Y denote the number of balls in the randomly selected bucket. Then Y is a 2 nonnegative random variable with E [Y ] = 2, so by Markov’s inequality, P {Y ≥ 5} ≤ 5 = 0.4. That is, the fraction of buckets with ﬁve or more balls is less than or equal to 0.4. Equality is achieved if and only if the only possible values of Y are zero and ﬁve, that is, if and only if each bucket is either empty or has exactly ﬁve balls. 46 CHAPTER 2. DISCRETE-TYPE RANDOM VARIABLES Second, the Chebychev inequality states that if X is a random variable with ﬁnite mean µ and variance σ 2 , then for any d &gt; 0, σ2 P {|X − µ| ≥ d} ≤ 2 . (2.9) d The Chebychev inequality follows by applying the Markov inequality with Y = |X − µ|2 and c = d2 . A slightly diﬀerent way to write the Chebychev inequality is to let d = aσ, for any constant a &gt; 0, to get 1 (2.10) P { X − µ| ≥ aσ } ≤ 2 . a In words, this form o...
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