Abstract
Poisson random variables are used to model the number of occurrences of certain events that come from a large number of independent sources, such as the number of calls to an office telephone during business hours, the number of atoms decaying in a sample of some radioactive substance, the number of visits to a website, or the number of customers entering a store.
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Notes
- 1.
Named after Simeon D. Poisson (1781–1840), who in 1837 introduced them to model the votes of jurors, although Abraham de Moivre had already considered them about a hundred years earlier.
- 2.
Any such assumption is always just an approximation that is usually valid only within three or four standard deviations from the mean. But that is the range where almost all of the probability of the normal distribution falls, and although theoretically the tails of the normal distribution are infinite, \(\varphi (z)\) is so small for | z | > 4, that as a practical matter we can ignore the fact that it gives nonzero probabilities to impossible events such as people having negative heights or heights over ten feet.
- 3.
De Moivre discovered the normal curve and proved this theorem only for \(p = \frac{1} {2}\). For \(p\neq \frac{1} {2}\) he only sketched the result. It was Pierre-Simon de Laplace who around 1812 gave the details for arbitrary p, and outlined a further generalization, the Central Limit Theorem, which we will discuss shortly.
- 4.
If \(n \rightarrow \infty\) with p and k fixed, then \(n - i + 1 \sim n - i\) for all i ≤ k and \(np \sim \left [np\right ]\), as well.
- 5.
By 7.3.1 \(200\overline{X}\) and \(200\overline{Y }\) are both approximately normal and they are also independent, hence their difference is also approximately normal.
- 6.
1μ g = 1 microgram = 10−6 g
- 7.
Some authors define a negative binomial r.v. as the number of failures before the rth success, rather than the total number of trials.
- 8.
Expand \(\left (1 + x\right )^{-r}\) in a Taylor series.
- 9.
We assume 00 = 1 where necessary.
- 10.
A quadratic form (that is, a polynomial with quadratic terms only) whose coefficients satisfy these conditions is called positive definite , because its values are then positive for any choice of x 1 and x 2. (See any linear algebra text.)
- 11.
A quadratic form is positive semidefinite if its value is ≥ 0 for all arguments. For instance, in two dimensions \(\left (x + y\right )^{2}\) is positive semidefinite.
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Schay, G. (2016). Some Special Distributions. In: Introduction to Probability with Statistical Applications. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-30620-9_7
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DOI: https://doi.org/10.1007/978-3-319-30620-9_7
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