Extending the applicability of the RVT technique for the randomized radioactive decay chain model

Abstract

In this paper, we improve the existing analysis on the randomized radioactive decay chain model based on Bateman master equations, by Hussein and Selim (2020). For a decay chain of three species of radionuclides, the authors derived the probability density function for the concentrations, by using the random variable transformation technique. We extend this application to the general solution of Bateman equations. The density function is expressed as an expectation, which has important implications for parametric density estimation. This may improve the classical kernel estimation when the random dimensionality is not low. Numerical examples are included, where the decay parameters and the initial concentrations are assigned different probability distributions.

Appendix

We prove (2). Let ${\mathcal {C}}$ be a Borel set in ${\mathbb {R}}$. By definition of conditional law,

$$\begin{aligned} {\mathbb {P}}(Z_1A+Z_2\in {\mathcal {C}})= {}&\int _{{\mathbb {R}}^2}{\mathbb {P}}\left( Z_1A+Z_2\in {\mathcal {C}}|Z_1=z_1,Z_2=z_2\right) {\mathbb {P}}_{(Z_1,Z_2)}\left( \mathrm {d} z_1,\mathrm {d} z_2\right) \\ = {}&\int _{{\mathbb {R}}^2}{\mathbb {P}}\left( z_1A+z_2\in {\mathcal {C}}\right) {\mathbb {P}}_{(Z_1,Z_2)}\left( \mathrm {d} z_1,\mathrm {d} z_2\right) . \end{aligned}$$

Since A has a PDF,

$$\begin{aligned} {\mathbb {P}}\left( Z_1A+Z_2\in {\mathcal {C}}\right) ={}&\int _{{\mathbb {R}}^2}\int _{\left( {\mathcal {C}}-z_2\right) /z_1} f_A(a)\,\mathrm {d} a\,{\mathbb {P}}_{(Z_1,Z_2)}\left( \mathrm {d} z_1,\mathrm {d} z_2\right) \\ = {}&\int _{{\mathbb {R}}^2}\int _{{\mathcal {C}}} f_A\left( \frac{a-z_2}{z_1}\right) \frac{1}{|z_1|}\,\mathrm {d} a\,{\mathbb {P}}_{(Z_1,Z_2)}\left( \mathrm {d} z_1,\mathrm {d} z_2\right) . \end{aligned}$$

By Fubini’s Theorem (justified because the integrand is nonnegative) and the expression of the expectation,

$$\begin{aligned} {\mathbb {P}}\left( Z_1A+Z_2\in {\mathcal {C}}\right) = {}&\int _{{\mathcal {C}}}\int _{{\mathbb {R}}^2}f_A\left( \frac{a-z_2}{z_1}\right) \frac{1}{|z_1|}{\mathbb {P}}_{(Z_1,Z_2)}\left( \mathrm {d} z_1,\mathrm {d} z_2\right) \,\mathrm {d} a \\ = {}&\int _{{\mathcal {C}}} {\mathbb {E}}\left[ f_A\left( \frac{a-Z_2}{Z_1}\right) \frac{1}{|Z_1|}\right] \,\mathrm {d} a. \end{aligned}$$

This proves (2).

Though not used in the current paper, (2) may be generalized as follows. Let $\pmb {A}$ be an absolutely continuous random vector of length m. Let $\pmb {M}$ be an $m\times m$ random matrix and $\pmb {Z}$ be a random vector of length m. Suppose that $\pmb {A}$ is independent of $(\pmb {M},\pmb {Z})$ and that $\pmb {M}$ is invertible almost surely. Then, $\pmb {M}\pmb {A}+\pmb {Z}$ is absolutely continuous, with PDF

$$\begin{aligned} f_{\pmb {M}\pmb {A}+\pmb {Z}}(\pmb {a})={\mathbb {E}}\left[ f_{\pmb {A}}\left( \pmb {M}^{-1} (\pmb {a}-\pmb {Z})\right) \frac{1}{|\det (\pmb {M})|}\right] . \end{aligned}$$