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Unformatted text preview: as an approximately Gaussian distribution.
That is, if the sum is X , and if X is a Gaussian random variable with the same mean and variance
as X , then X and X have approximately the same CDF:
P {X ≤ v } ≈ P {X ≤ v } (Gaussian approximation). An important special case is when X is the sum of n Bernoulli random variables, each having the
same parameter p. In other words, when X has the binomial distribution, with parameters n and p.
The approximation is most accurate if both np and n(1 − p) are at least moderately large, and the
probabilities being approximated are not extremely close to zero or one. As an example, suppose
X has the binomial distribution with parameters n = 10 and p = 0.2. Then E [X ] = np = 2 and
Var(X ) = np(1 − p) = 1.6. Let X be a Gaussian random variable with the same mean and variance
as X. The CDFs of X and X are shown in Figure 3.11. The two CDFs cannot be close everywhere Figure 3.11: The CDF of a binomial random variable with parameters n = 10 and p = 0.2, and the
Gaussian approximation of it.
because the CDF of X is piecewise constant with jumps at integer points, and the CDF of X is
continuous. Notice, however, that the functions are particularly close when the argument is halfway
between two consecuti...
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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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