Abstract
We will explore now the protocols used to define on \((\Omega ,\mathcal{F})\) a probability \({\varvec{P}}\) also called either law or distribution , and we will start with finite or countable spaces so that \({\varvec{P}}\) will be defined in an elementary way.
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- 1.
A matrix \({\mathbb A}\) is non-negative definite if, however taken a vector of real numbers \(\varvec{x}=(x_1,\ldots ,x_n)\), it is always
$$\begin{aligned} \varvec{x}\cdot \mathbb {A}\varvec{x}=\sum _{i,j=1}^n a_{ij}x_ix_j\ge 0 \end{aligned}$$and it is positive definite if this sum is always strictly positive (namely non zero). If \({\mathbb A}\) is positive, it is also non singular, namely its determinant \(|{\mathbb A}|>0\) does not vanish, and hence it has an inverse \({\mathbb A}^{-1}\).
- 2.
Since the laws with the pdf’s f and g are ac, the boundaries of the chosen domains have zero measure, and hence we can always take such domains as closed without risk of errors.
References
Shiryaev, A.N.: Probability. Springer, New York (1996)
Métivier, M.: Notions Fondamentales de la Théorie des Probabilités. Dunod, Paris (1972)
Nelsen, R.B.: An Introduction to Copulas. Springer, New York (1999)
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Cufaro Petroni, N. (2020). Distributions. In: Probability and Stochastic Processes for Physicists. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-48408-8_2
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DOI: https://doi.org/10.1007/978-3-030-48408-8_2
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