98 Maˆ ıtriser l’al´ eatoire
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z
1-z
X+Y-7=z
X+Y-7=0
A 1
A 2
Fig. 4.4 – Agrandissement de l’aire hachur´ ee de la figure 4.3 de
l’exercice 4.8.
l’aire hachur´ ee horizontalement (A 1 ) sur l’aire du triangle (A T = A 1 + A 2 ) et
on obtient
P (X + Y − 7 ≤ z) =
aire A 1
aire A T
=
aire A T − aire A 2
aire A T
=
1
2 −
1
2 (1 − z)
2
1
2
= 1 − (1 − z)
2 = 2z − z
2 .
4.9
Pour obtenir la probabilit´ e conditionnelle de X | Y , il faut calculer la densit´ e
marginale de Y
f Y (y) =
∞
0
1
y
exp
−
x
y
− y
dx =
e
−y
y
−y exp
−
x
y
x=∞
x=0
= e
−y , y > 0
et ainsi obtenir la densit´ e conditionnelle
f X|Y =y (x) =
f X,Y (x, y)
f Y (y)
=
1
y
exp
−
x
y
, x > 0, y > 0.
La probabilit´ e cherch´ ee est donc
P (X > 1 | Y = y) =
∞
1
1
y
exp
−
x
y
dx = exp
−
1
y
.