Abstract
Normally distributed data spawn a set of fundamentally important theoretical distribution densities.
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Notes
- 1.
As has already been annotated, we identify random variables by upper-case letters and their realizations by lower-case letters. In case of Greek letters, however, we shall distinguish random variables and their realizations simply via context.
- 2.
Consider the sums \(\sum_{l=1}^{n}(x_{l}-\mu_{x})\) and \(\sum_{l=1}^{n}(x_{l}-\bar{x})\). While the first need not be zero, the second must. Obviously, in the latter n−1 differences settle the nth. From there, the first has ν=n and the latter ν=n−1 degrees of freedom [55].
- 3.
Even if the X i ; i=1,…,m are mutually dependent, the realizations z (l), l=1,…,n of the random variable Z are independent. This is obvious as the realizations \(x_{i}^{(l)}\) of X i , with i fixed and l running, are independent.
- 4.
As an exception, here we deviate from the nomenclature agreed on according to which f shall denote a systematic error.
- 5.
Let f X (x) denote the density of a continuous random variable X. Given a probability P, x satisfying \(\int_{0}^{x} f_{X}(x')\; dx'=P\) is termed a quantile of order P.
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Grabe, M. (2014). Normal Parent Distributions. In: Measurement Uncertainties in Science and Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-04888-8_3
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DOI: https://doi.org/10.1007/978-3-319-04888-8_3
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