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6.6 On suppose que 0 ≤ x ≤ 1<strong>et</strong>0≤y ≤ 1. Toutes les densités sont nulles en dehors de<br />
ce rectangle.<br />
(a)<br />
x y<br />
FXY = P (X ≤ x, Y ≤ y) = fXY (u, v)dudv<br />
x y<br />
0 0<br />
= (2u +2v − 4uv)dudv<br />
(b) Densité marginale de X:<br />
Densité marginale de Y :<br />
0<br />
x<br />
0<br />
=<br />
0<br />
(2uv + v 2 − 2uv 2 )| y<br />
0du =(u 2 y + uy 2 − u 2 y 2 )| x 0 = xy(x + y − xy).<br />
+∞<br />
fX(x) = fXY (x, v)dv<br />
−∞<br />
1<br />
= (2x +2v − 4xv)dv<br />
0<br />
=2x +[v 2 ] 1 0 − 2x[v2 ] 1 0<br />
=1.<br />
+∞<br />
fY (y) = fXY (u, y)du<br />
−∞<br />
1<br />
= (2u +2y − 4uy)du<br />
0<br />
=[u 2 ] 1 0 +2y[u] 1 0 − y[2u 2 ] 1 0<br />
=1.<br />
(c) fX|Y =y(x|y) =fXY (x, y)/fY (y) =2x +2y − 4xy/1 =2x +2y − 4xy.<br />
fY |X=x(y|x) =fXY (x, y)/fX(x) =2x +2y − 4xy/1 =2x +2y − 4xy.<br />
(d) X <strong>et</strong> Y ne sont pas indépendants car fXY (x, y) =2x+2y−4xy = 1·1 =fX(x)·fY (y).
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