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Unformatted text preview: inomial distributions have two parameters, namely n and p, and they involve binomial coefﬁcients, which can be cumbersome. Poisson distributions are simpler–having only one parameter,
λ, and no binomial coeﬃcients. So it is worthwhile using the Poisson distribution rather than the
binomial distribution for large n and small p. We now derive a limit result to give evidence that
this is a good approximation. Let λ > 0, let n and k be integers with n ≥ λ and 0 ≤ k ≤ n, and let
pb (k ) denote the probability mass at k of the binomial distribution with parameters n and p = λ/n.
We ﬁrst consider the limit of the mass of the binomial distribution at k = 0. Note that
ln pb (0) = ln(1 − p)n = n ln(1 − p) = n ln 1 − λ
n . λ
By Taylor’s theorem, ln(1 + u) = u + o(u) where o(u)/u → 0 as u → 0. So, using u = − n , ln pb (0) = n − λ
n → −λ as n → ∞. Therefore,
pb (0) = 1− λ
n n → e−λ as n → ∞, (2.5) 42 CHAPTER 2. DISCRETE-TYPE...
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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
- The Land