A variational problem to calculate probabilities
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909547098013696 |
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| author | Reyna-Castañeda, Hugo Guadalupe Romero, María de los Ángeles Sandoval |
| author_facet | Reyna-Castañeda, Hugo Guadalupe Romero, María de los Ángeles Sandoval |
| contents | In this paper, we prove the existence and uniqueness of the conditional expectation of an event $A$ given a $σ$-algebra $\mathcal{G}$ as a linear problem in the Lebesgue spaces $L^{p}$ associated with a probability space through the Riesz Representation Theorems. For the $L^{2}$ case, we state the Dirichlet's principle. Then, we extend this principle for specific values of $p$, framing the existence of the conditional expectation as a variational problem. We conclude with a proof of the law of total probability using these tools. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_16727 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A variational problem to calculate probabilities Reyna-Castañeda, Hugo Guadalupe Romero, María de los Ángeles Sandoval Probability Functional Analysis Optimization and Control 60A05, 60A10, 46E15, 46E20, 49K35, 49J50 In this paper, we prove the existence and uniqueness of the conditional expectation of an event $A$ given a $σ$-algebra $\mathcal{G}$ as a linear problem in the Lebesgue spaces $L^{p}$ associated with a probability space through the Riesz Representation Theorems. For the $L^{2}$ case, we state the Dirichlet's principle. Then, we extend this principle for specific values of $p$, framing the existence of the conditional expectation as a variational problem. We conclude with a proof of the law of total probability using these tools. |
| title | A variational problem to calculate probabilities |
| topic | Probability Functional Analysis Optimization and Control 60A05, 60A10, 46E15, 46E20, 49K35, 49J50 |
| url | https://arxiv.org/abs/2503.16727 |