51 05 16 641 001 ml parameter estimation for

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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 (DeMoivre-Laplace 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 DeMoivre-Laplace 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.

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