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L p = e (z - x)! x! Z! l 1 Z! (/1, + p) Z On montre ainsi la propriete suivante : Si X et Y sont deux variables de Poisson de parametres A. et Jl, independantes, leur somme X + Y suit une loi de Poisson de parametre A. + Jl. V-1-2 Cas general V-J-2-1. Theoreme La loi de probabilite de la somme Z de deux variables independantes est la me sure image de Px Q9 Py par l'application : (x , y)~ x + y de R2 dans R. C'est Ie produit de convolution des deux mesures. Pour tout bore lien B : PZ(B) = Sf XB(x+ y)dPx(x)dPy(y) R2 44 Ch.

J VII-l-l Loi conjointe VIJ-J-J-J. Definition La loi du couple (X, Y), appelee loi conjointe, est definie par la donnee des nombres Pi) VIJ-J-J-2. / I,I,Pij i j =1 Proprietes 58 Ch. VII • Couple de variables aleatoires VII-I-2 Lois marginales On appelle loi rnarginale de X, la loi de X prise separernent : q P(X = xi) = LPij = Pi. j=l On detinit, de rnerne, la loi rnarginale de Y. VII -1-3 Lois conditionnelles Considerons deux evenernents {X = Xi} et {Y = yj} de probabilites non nulles. 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.

Sa courbe representative est la suivante : IV-5-3-2. Courbe representative de la [onetion de repartition F(t) 38 Ch. / Les moments d' ordre impair k = 2n+ 1 sont nuls (la fonction integree est impaire) . J2n R+ 2 dt = udu = J2idu On pose t=2 1 f n-"2 exp()d 2 n r( 1) 2 n 1x3x5 ... Jn 2n! 112n =-n2 n! Dans Ie cas d'une variable norrnale quelconque on a : • Le coefficient d' asymetrie (Skewness) : • Le coefficient d'aplatissement (Kurtosis) : 112n (2n)! = - -n (,J 2 n. 2n Ch. IV • Lois de probabilite continues 39 IV -5-6 Theoreme Soit X et Y deux variables aleatoires independantes suivant les lois LG( mba 1 ) et LG(m2,a2), alors la variable aleatoire X+Y suit la loi normale : r LG(ml +m2,Ja +ai).

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