Unformatted text preview: binomial random variable with parameters n = 30 and p = 0.2, and the
Gaussian approximation of it.
As mentioned earlier, the Gaussian approximation is backed by various forms of the central limit
theorem (CLT). Historically, the ﬁrst version of the CLT proved is the following theorem, which
pertains to the case of binomial distributions. Recall that if n is a positive integer and 0 < p < 1,
a binomial random variable Sn,p with parameters n and p has mean np and variance np(1 − p).
S −np
.
Therefore, the standardized version of Sn,p is √n,p
np(1−p) Theorem 3.6.7 (DeMoivreLaplace limit theorem) Suppose Sn,p is a binomial random variable
with parameters n and p. For p ﬁxed, with 0 < p < 1, and any constant c,
lim P n→∞ Sn,p − np
np(1 − p) ≤c = Φ(c). 3.6. LINEAR SCALING OF PDFS AND THE GAUSSIAN DISTRIBUTION 95 The practical implication of the DeMoivreLaplace limit theorem is that, for large n, the standardized version of Sn,p has approximately the standard normal distribution, or equivalently, that Sn,p
has approximately the same CDF as a Gaussian random v...
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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 Tech.
 Spring '08
 Zahrn
 The Land

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