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Unformatted text preview: ariable with the same mean and variance.
That is, the DeMoivreLaplace limit theorem gives evidence that the Gaussian approximation to
the binomial distribution is a good one. A more general version of the CLT is stated in Section
4.9.2.
Example 3.6.8 (a) Suppose a fair coin is ﬂipped a thousand times, and let X denote the number
of times heads shows. Using the Gaussian approximation with the continuity correction,2 ﬁnd the
approximate numerical value K so P {X ≥ K } ≈ 0.01. (b) Repeat, but now assume the coin is
ﬂipped a million times.
Solution: (a) The random variable X has the binomial distribution with parameters n = 1000 and
√
p = 0.5. It thus has mean µX = np = 500 and standard deviation σ = np(1 − p) = 250 ≈ 15.8.
By the Gaussian approximation with the continuity correction,
P {X ≥ K } = P {X ≥ K − 0.5} = P X −µ
K − 0.5 − µ
≥
σ
σ ≈Q K − 0.5 − µ
σ . Since Q(2.325) ≈ 0.01 we thus want to select K so K −0.5−µ ≈ 2.325 or K = µ + 2.325σ + 0...
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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
 Zahrn
 The Land

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