A Bayesian model for local smoothing in kernel density by Brewer M. J.

By Brewer M. J.

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Dans ce chapitre, nous allons exposer les generalites concernant un couple de variables aleatoires. Nous donnerons ensuite quelques resultats essentiels concernant les vecteurs aleatoires. Usuellement, la loi de probabilite d'un couple (X, Y) de variables aleatoires reelles est une application de l'espace probabilise (n, C, P) dans R2 muni de sa tribu de Borel. Toutefois dans la premiere partie de ce chapitre, nous allons nous interesser a deux variables prenant leurs valeurs dans des ensembles finis ou denombrables.

On peut alors definir deux lois conditionnelles en rappelant que X et Y peuvent etre des variables qualitatives (cf chapitre I) : Loi conditionnelle de X sachant Y = l:i VII-1-3-1. P(X VII-1-3-2. } = y} /X = xi) P(X = = Xi n Y = y}) P(X = Xi) = Pi} Pi. Remarques II est frequent d'utiliser la connaissance que l'on a d'une loi conditionnelle et de la loi rnarginale correspondante pour determiner la loi du couple, ceci a l'aide du theorerne des probabilites totales : P(X • /Y Loi conditionnelle de Y sachant X = xi P(Y VII-1-3-3.

A-,=np>18 ... ,. Loi de Laplace-Gauss LG(np,~np(l- p)) 56 Ch. VI • Convergences de suites de variables Ch. VII • Couple de variables aleatoires Ch. VII 57 Couple de variables aleatoires Dans Ie chapitre I, nous avons etudie Ie conditionnement d'un evenement par un autre evenement. Plus generalement, beaucoup d'etudes statistiques metient en jeu plusieurs variables aleatoires et nous nous interessons aleurs lois conjointes, conditionnelles ... Dans ce chapitre, nous allons exposer les generalites concernant un couple de variables aleatoires.

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