Lois de probabilité issues de gaussiennes réitérées

International audienceOn considère un échantillon aléatoire $(X_1,...,X_n)$ suivant la loi normale $N(mu,sigma_1^2)$, de taille $n> 1$. Conditionnellement à chaque $X_i$, i=1,...,n, on définit un nouvel échantillon aléatoire $(X_{i,1},...,X_{i,n})$ suivant la loi normale $N(X_i,\sigma_2^2)$ $(N(X_i,\sigma_2^2)$ est une notation introduite par commodité). Sous l'hypothèse que les n nouveaux échantillons aléatoires ainsi obtenus sont conditionnellement indépendants, on obtient un ensemble de points aléatoires de seconde génération. La question est d'étudier les propriétés de cet ensemble. On donne un théorème précisant la densité limite obtenue lorsque n tend vers l'infini, et on généralise ce théorème en étudiant ce qui se produit lorsque que l'on répète cette procédure jusqu'à obtenir, conditionnellement à chaque $X_{i_1,i_2,...,i_{p-1}}$, $i_1=1,..., n_1,i_2=1,...,n_2,..., i_{p-1}=1,...,n_{p-1},$ de nouveaux échantillons aléatoires $X_{i_1,i_2,...,i_p}, i_p=1,..., n_p$ suivant la loi normale $N(X_{i_1,i_2,...,i_{p-1}},\sigma_p^2)$

New Concepts in the Evaluation of Biodegradation/Persistence of Chemical Substances Using a Microbial Inoculum

The European REACH Regulation (Registration, Evaluation, Authorization of CHemical substances) implies, among other things, the evaluation of the biodegradability of chemical substances produced by industry. A large set of test methods is available including detailed information on the appropriate conditions for testing. However, the inoculum used for these tests constitutes a “black box.” If biodegradation is achievable from the growth of a small group of specific microbial species with the substance as the only carbon source, the result of the test depends largely on the cell density of this group at “time zero.” If these species are relatively rare in an inoculum that is normally used, the likelihood of inoculating a test with sufficient specific cells becomes a matter of probability. Normally this probability increases with total cell density and with the diversity of species in the inoculum. Furthermore the history of the inoculum, e.g., a possible pre-exposure to the test substance or similar substances will have a significant influence on the probability. A high probability can be expected for substances that are widely used and regularly released into the environment, whereas a low probability can be expected for new xenobiotic substances that have not yet been released into the environment. Be that as it may, once the inoculum sample contains sufficient specific degraders, the performance of the biodegradation will follow a typical S shaped growth curve which depends on the specific growth rate under laboratory conditions, the so called F/M ratio (ratio between food and biomass) and the more or less toxic recalcitrant, but possible, metabolites. Normally regulators require the evaluation of the growth curve using a simple approach such as half-time. Unfortunately probability and biodegradation half-time are very often confused. As the half-time values reflect laboratory conditions which are quite different from environmental conditions (after a substance is released), these values should not be used to quantify and predict environmental behavior. The probability value could be of much greater benefit for predictions under realistic conditions. The main issue in the evaluation of probability is that the result is not based on a single inoculum from an environmental sample, but on a variety of samples. These samples can be representative of regional or local areas, climate regions, water types, and history, e.g., pristine or polluted. The above concept has provided us with a new approach, namely “Probabio.” With this approach, persistence is not only regarded as a simple intrinsic property of a substance, but also as the capability of various environmental samples to degrade a substance under realistic exposure conditions and F/M ratio

FINDING NEW LIMIT POINTS OF MAHLER MEASURE BY METHODS OF MISSING DATA RESTORATION

It is well known that the set of Mahler measures of single variable polynomial has limit points of which a list established by D. Boyd and M. Mossinghoff has been extended through approaches based on genetic algorithms. In this paper, we wish to further extend the list of known limit points by adapting a method of missing data restoration

Modèles de distribution et stratégies d'échantillonnage des populations bactériennes dans un réseau de distribution publique

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Etude des méthodes de quantification en microbiologie

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Définition d'une stratégie de prélèvement pour l'analyse microbiologique des eaux de surface

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Heterogeneity: A Major Factor Influencing Microbial Exposure and Risk Assessment

International audienceMicrobial risk assessment is dependent on several biological and environmental factors that affect both the exposure characteristics to the biological agents and the mechanisms of pathogenicity involved in the pathogen-host relationship. Many exposure assessment studies still focus on the location parameters of the probability distribution representing the concentration of the pathogens and/or toxin. However, the mean or median by themselves are insufficient to evaluate the adverse effects that are associated with a given level of exposure. Therefore, the effects on the risk of disease of a number of factors, including the shape parameters characterizing the distribution patterns of the pathogen in their environment, were investigated. The statistical models, which were developed to provide a better understanding of the factors influencing the risk, highlight the role of heterogeneity and its consequences on the commonly used risk assessment paradigm. Indeed, the heterogeneity characterizing the spatial and temporal distribution of the pathogen and/or the toxin contained in the water or food consumed is shown to be a major factor that may influence the magnitude of the risk dramatically. In general, the risk diminishes with higher levels of heterogeneity. This scheme is totally inverted in the presence of a threshold in the dose-response relationship, since heterogeneity will then have a tremendous impact, namely, by magnifying the risk when the mean concentration of pathogens is below the threshold. Moreover, the approach of this article may be useful for risk ranking analysis, regarding different exposure conditions, and may also lead to improved water and food quality guidelines

Etude des méthodes de quantification en microbiologie

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Probability distributions arising from nested Gaussians

International audienceWe consider a random sample X1,...,Xn of size n⩾1 from an View the MathML source Gaussian law. Then, conditionally on each View the MathML source, we define a new random sample Xi,1,...,Xi,n from the View the MathML source normal distribution (View the MathML source is notation introduced for convenience). Assuming that the so obtained n new random samples are conditionally independent, we get a second step randomly generated set of points. The question is to investigate the properties of this set. We give a theorem precising the limiting density obtained when n approaches infinity, and we generalize this theorem by studying what occurs when repeating this process until, conditionally on each View the MathML source, we get new random samples View the MathML source, from the View the MathML source normal distributio