3. Variables al´ eatoires continues 59
3.3
1. Si 0 < x < π/2
F X (x) =
x
0
sin(t)dt = − cos(t)
t=x
t=0
= 1 − cos(x).
Par cons´ equent
F X (x) =



0
s ix ≤ 0
1 − cos(x) si 0 < x < π/2
1
s i n o n .
2. L’esp´ erance de X est
E(X) =
π/2
0
x sin(x)dx =
= − (x cos(x))
π/2
0
=0
+
π/2
0
cos(x)dx = sin(x)
x=π/2
x=0
= 1.
3. Calculons le 2
e moment de X
E(X
2 ) =
π/2
0
x
2 sin(x)dx =
= −
x
2 cos(x)
π/2
0
=0
+2
π/2
0
x cos(x)dx
= 2(x sin(x))
π/2
0
=π
−2
π/2
0
sin(x)dx = π − 2,
et la variance vaut
var(X) = E(X
2 ) − E(X)
2 = π − 3.
3.4
1. La fonction de r´ epartition de X pour x ≥ 0 est
F X (x) =
t
0
λe
−λt dt = −e
−λt
t=x
t=0
= 1 − e
−λx ,
et donc
F X (x) =
0
s ix < 0
1 − e
−λx si x ≥ 0.