Corrigé des exercices - Dunod

372
Exercice 30.19
1. Par définition, m k = ∑ x k P(X = x) = ∑ − E(X) + E(X))
x∈I
x∈I(x k P(X = x).
m k = ∑ x∈I
k∑
=
=
i=0
k∑
i=0
k∑
i=0
( k
i)
(x − E(X)) i E(X) k−i P(X = x)
( k
i)
E(X) k−i ∑ x∈I(x − E(X)) i P(X = x)
( k
i)
µ i E(X) k−i
µ k = ∑ x∈I(x − E(X)) k P(X = x)
= ∑ x∈I
=
=
2. Remarquons que m 1 = µ [1] .
k∑
i=0
k∑
i=0
k∑
i=0
( k
i)
x i (−E(X)) k−i P(X = x)
( k
i)
(−E(X)) k−i ∑ x∈I
x i P(X = x)
( k
i)
m i (−E(X)) k−i
m 2 = ∑ x∈I
x 2 P(X = x)
= ∑ x∈I
= ∑ x∈I
(x(x − 1) + x)P(X = x)
x(x − 1)P(X = x) + ∑ x∈I
xP(X = x)
= µ [2] + µ [1]
m 3 = ∑ x 3 P(X = x)
x∈I
= ∑ (x(x − 1)(x − 2) + 3x(x − 1) + x)P(X = x)
x∈I
= ∑ x(x − 1)(x − 2)P(X = x) + 3 ∑ x(x − 1)P(X = x) + ∑ xP(X = x)
x∈I
x∈I
x∈I
3.
= µ [3] + 3µ [2] + µ [1]