Abstract
Consider the problem of constructing a model of the process (experiment) of throwing a die and observing the outcome; in doing so, we need to impose on the experiment a certain probabilistic framework since the same die thrown under ostensibly identical circumstances, generally, yields different outcomes. The framework represents, primarily, the investigator’s view of the nature of the process, but it must also conform to certain logical rules.
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Notes
- 1.
Certain other usages are also common; thus the collection \(\mathcal{J}\) is also denoted by \(\mathcal{J} = \mathcal{A}_{1} \times \mathcal{A}_{2}\), which is to be distinguished from \(\mathcal{A}_{1} \otimes \mathcal{A}_{2}\), the latter being equal to \(\sigma (\mathcal{J} )\),
- 2.
In this argument it is assumed that the collections \(Z_{\mathcal{C}},\ Z_{\mathcal{D}}\), are nonempty; otherwise there is nothing to prove. Evidently, if the collections \(\mathcal{C},\ \mathcal{D}\) are algebras then it is easy to see that \(Z_{\mathcal{C}},\ Z_{\mathcal{D}}\) are nonempty collections.
- 3.
This is a consequence of the fact that if we have two (collections of) sets obeying \(\mathcal{C}_{1} \subset \mathcal{C}_{2}\) then the σ-algebras they generate obey \(\sigma (\mathcal{C}_{1}) \subset \sigma (\mathcal{C}_{2})\).
- 4.
When the context is clear and no confusion is likely to arise we shall generally use the notation E  X.
- 5.
In order to avoid this cumbersome phraseology in the future, when we say that X is a random variable, it is to be understood that we have predefined the appropriate probability and measurable spaces. Thus, mention of them will be suppressed.
- 6.
It is a common practice that the subscript for the first moment is omitted for both scalar and vector random variables. We shall follow this practice unless reasons of clarity require otherwise.
- 7.
For twice differentiable convex functions the matrix of the second order partial derivatives is positive semidefinite; for concave functions, it is negative semidefinite; in both cases this is to be understood in an a.e. sense.
- 8.
This result is obtained by using a Taylor series expansion around the point x 0, retaining terms up to and including the second derivative (third term of the expansion).
- 9.
Note that Ω j , j = 1, 2, are exact copies of the same space.
- 10.
To connect this with the earlier discussion of the topic note that
$$\displaystyle{P(D_{k}\mid D_{j2}) = \frac{P(D_{k} \cap D_{j2})} {P(D_{j2})} = P(D_{k-j,1} \cap D_{j2})/P(D_{j2}).}$$ - 11.
These statements are to be understood in the a.c. sense, where appropriate.
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Dhrymes, P.J. (2013). Foundations of Probability. In: Mathematics for Econometrics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8145-4_8
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