Fonction quantile Fichier

F µ R
µ F G
]0,1[ R
∀α∈]0,1[ G(α) = inf{t∈R:F(t)≥α}.
F µ
RG µ F
G
−2 0 2 4 6
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x
y
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0.0 0.2 0.4 0.6 0.8 1.0
−2 0 2 4 6
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y
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G
∀α∈]0,1[ F[G(α)] ≥α F [G(α)−]≤α
F∀α∈]0,1[ F[G(α)] = α

G n ≥1F[G(α) + 1
n]≥α F [G(α)−1 n]< α
G α ∈]0,1[ b= lim
n→∞G(α−1
n)≤G(α)
F[G(α−1
n)] ≥α−1 nF(b)≥α F (b−)≥α
F(b)≥α G(α)≤b G(α) = lim
n→∞G(α−1
n)
G
G(α){F≥α}
α≤F(t)⇐⇒ G(α)≤t
F(t)< α ⇐⇒ t < G(α)
t{G≤t}=]0, F (t)] F(t)<1{G≤t}=]0,1] F(t) = 1
U(Ω,F, P ) ]0,1[ U0,1
G◦U µ G
]0,1[ µ
{G◦U≤t}={ω∈Ω : G[U(ω)] ≤t}={ω∈Ω : U(ω)≤F(t)}={U≤F(t)},
P(G◦U≤t) = P(U≤F(t)) = F(t)U
]0,1[ G◦U µ
G◦U µ
FR R
lim
x→−∞F(x) = 0 lim x→+∞F(x) = 1 µ
F

G F U
]0,1[ U0,1µ G ◦U
F
FR R
[u, v]µRx y
x≤y µ(]x, y]) = F(y)−F(x)
a= lim
x→−∞F(x)b= lim x→+∞F(x)
a=b F = 0
a6=bF−a
b−aµ
RF−a
b−aν= (b−a)µ x ≤y∈R
ν(]x, y]) = (b−a)µ(]x, y]) = (b−a)hF(y)−a
b−a−F(x)−a b−ai=F(y)−F(x).
X(Ω,F, P )µ
F F F ◦XU0,1
lim
x→−∞F(x) = 0 lim x→+∞F(x)=1 x∈]0,1[ {F=x}
F[a, b]a≤b b F (b) = x
{F≤x}=] − ∞, b]
{F◦X≤x}={X≤b}P(F◦X≤x) = P(X≤b) = F(b) = x,
F◦XU0,1