Abstract
The problem introduced in Sect. 1.6.1 with Bertrand’s paradox occurs when we try to decompose the range of possible values of a random variable x into equally probable elementary intervals and this is not always possible without ambiguity because of the continuous nature of the problem. In Sect. 1.6 we considered a continuous random variable x with possible values in an interval [x 1, x 2], and we saw that if x is uniformly distributed in [x 1, x 2], a transformed variable y = Y (x) is not in general uniformly distributed in [y 1, y 2] = [Y (x 1), Y (x 2)] (Y is taken as a monotonic function of x). This makes the choice of the continuous variable on which equally probable intervals are defined an arbitrary choice.
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Notes
- 1.
If we assume here that f(x) is sufficiently regular, such that \(P(\{\tilde{x}\}) = 0\), i.e. f(x) has no Dirac’s delta component \(\delta (x -\tilde{ x})\), we have \(P(x <\tilde{ x}) = P(x>\tilde{ x}) = 1/2\). Otherwise, \(P(x <\tilde{ x}) = P(x>\ \tilde{x})\ = (1 - P(\{\tilde{x}\})\,)/2\).
- 2.
Γ(n) = (n − 1)! if n is an integer value.
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Lista, L. (2017). Probability Distribution Functions. In: Statistical Methods for Data Analysis in Particle Physics. Lecture Notes in Physics, vol 941. Springer, Cham. https://doi.org/10.1007/978-3-319-62840-0_2
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