Abstract
This contribution provides a practical view to the computation of the first probability density function of the solution stochastic process to ordinary and partial differential equations with randomness using the Random Variable Transformation technique. The analysis is performed via a set of simple examples, belonging to different areas like Physics, Biology and Engineering, with the aim of illustrating key ideas from a practical standpoint.
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Notes
- 1.
The Whishart distribution is a probability distribution for random matrices. A random matrix is a matrix whose entries are random variables. We say that a k × k random matrix A is absolutely continuous (respectively discrete) if its vectorization, vec(A), is an absolutely continuous (respectively discrete) k 2 × 1 random vector. In this case, the density or mass function of A is defined as the density or mass function of vec(A). We say that a k × k random matrix A follows a Wishart distribution, A ∼Wishart(H, n), H being k × k symmetric and positive definite matrix and n ≥ k, if it is symmetric, positive definite and absolutely continuous, with density
$$\displaystyle \begin{aligned} P_A(a)=c[\det(a)]^{\frac{n}{2}-\frac{k+1}{n}}e^{-\frac{n}{2}\text{tr}(H^{-1}a)}, \end{aligned}$$being c a constant depending on H and n.
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Calatayud, J., Cortés, JC., Jornet, M., Navarro-Quiles, A. (2019). Solving Random Ordinary and Partial Differential Equations Through the Probability Density Function: Theory and Computing with Applications. In: Sadovnichiy, V., Zgurovsky, M. (eds) Modern Mathematics and Mechanics. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-96755-4_15
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