Distribuições de Probabilidade
Alfredo D. Egídio dos Reis
ISEG, Abril de 1999
Conteúdo
1 Distribuições Discretas
1.1 Bernoulli (p) . . . . . .
1.2 Binomial (n, p) . . . . .
1.3 Binomial Negativa (k, p)
1.4 Geométrica(p) . . . . . .
1.5 Hipergeométrica(m, n, k)
1.6 Poisson (λ) . . . . . . .
1.7 Uniforme discreta (n) .
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1
1
1
1
1
1
1
2
2 Distribuições Contínuas
2.1 Beta (α, β) . . . . . . . . . . . . . . . .
2.2 Cauchy (α, β) . . . . . . . . . . . . . . .
2.3 Exponencial (β) . . . . . . . . . . . . . .
2.3.1 Combinação de Exponenciais . .
2.4 Exponencial Dupla ou de Laplace (α, β)
2.5 F -Snedcor, F (m, n) . . . . . . . . . . .
2.6 Gama (α, β) . . . . . . . . . . . . . . . .
2.7 Logística (α, β) . . . . . . . . . . . . . .
2.8 Gaussiana inversa (α, β) . . . . . . . . .
2.9 Lognormal (µ, σ 2 ) . . . . . . . . . . . .
2.10 Normal (µ, σ) . . . . . . . . . . . . . . .
2.11 Pareto (α, β) . . . . . . . . . . . . . . .
2.12 Pareto Generalizada (α, β, θ) . . . . . .
2.13 Qui-Quadrado χ2 (n) . . . . . . . . . . .
2.14 t-Student, t(n) . . . . . . . . . . . . . .
2.15 Uniforme (a, b) . . . . . . . . . . . . . .
2.16 Weibull (α, β) . . . . . . . . . . . . . . .
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i
1
Distribuições Discretas
1.1
Bernoulli (p)
f (x) = P r(X = x) = px (1 − p)1−x , x = 0, 1 (0 < p < 1)
E[X] = p, V [X] = p(1 − p), E[etX ] = (1 − p) + pet
Notas: Bernoulli(p) ≡ Binomial(1; p)
1.2
Binomial (n, p)
f (x) =
µ
n
x
¶
px (1 − p)n−x , x = 0, 1, 2, · · · , n (0 < p < 1; n = 1, 2, · · · )
£
¤n
E[X] = np, V [X] = np(1 − p), E[etX ] = pet + 1 − p
Notas: Binomial(1; p) ≡ Bernoulli(p)
1.3
Binomial Negativa (k, p)
f (x) =
µ
k+x−1
x
¶
pk (1 − p)x , x = 0, 1, 2, · · · (k = 1, 2, · · · , 0 < p < 1
k(1 − p)
k(1 − p)
E[etX ] =
E[X] =
V [X] =
p
p2
µ
p
1 − (1 − p)et
¶k
, t < − log(1 − p)
Notas: BN(1, p)≡Geométrica(p)
1.4
Geométrica(p)
f (x) = p(1 − p)x , x = 0, 1, 2, · · · (0 < p < 1)
x
X
F (x) = p
(1 − p)i = 1 − (1 − p)x+1 x = 0, 1, 2, · · ·
i=0
E[X] =
(1 − p)
p
(1 − p)
E[etX ] =
, t < − log(1 − p)
V [X] =
p
p2
1 − (1 − p)et
Notas: Geom(p)≡BN(1, p)
1.5
Hipergeométrica(m, n, k)
f (x) =
µ
m
x
¶µ
µ
n
k
n−m
k−x
¶
E[X] =
1.6
¶
, x = 0, 1, 2, · · · , k ,
µ
m−n+k ≤x≤m
n, m, k ≥ 0
KM
KM (N − M )(N − K)
, V [X] =
N
N
N (N − 1)
Poisson (λ)
λx
, x = 0, 1, 2, · · · (λ > 0)
x!
t
E[X] = V [X] = λ E[etX ] = eλ(e −1)
f (x) = e−λ
1
¶
1.7
Uniforme discreta (n)
1
, x = 1, 2, · · · , n (n = 1, 2, · · · )
n
x
, x = 0, 1, 2, · · · , n
n
f (x) =
F (x) =
n
X1
et (1 − ent )
(n2 − 1)
n+1
V [X] =
E[etX ] =
ekt =
2
12
n
n(1 − et )
E[X] =
k=1
2
2.1
Distribuições Contínuas
Beta (α, β)
f (x) = B (α, β)−1 xα−1 (1 − x)β−1 , 0 < x < 1 (α, β > 0)
Z 1
Γ(α)Γ(β)
β−1
xα−1 (1 − x)
dx =
B (α, β) =
Γ(α + β)
0
E[X] =
2.2
B(α + k, β)
α
αβ
E[X k ] =
V [X] =
2
α+β
B(α, β)
(α + β) (α + β + 1)
Cauchy (α, β)
f (x) =
α
³
´ , − ∞ < x < +∞ (α > 0, − ∞ < β < +∞)
2
2
π α + (x − β)
E[X] : não existe V [X] : não existe E[etX ] : não existe
Notas: Cauchy(1; 0) ≡ t(1)
y
0.08
0.07
0.06
0.05
0.04
0.03
-6
-4
-2
0
2
4
Densidade da Cauchy(2; 0)
2.3
Exponencial (β)
f (x; β) = βe−β x , x > 0 (β > 0)
F (x) = 1 − e−β x
E[X] =
1
β
1
V [X] = 2 E[etX ] =
β
β
−t
β
Notas: Exp(β) ≡ Gama(1, β)
2
6
x
y
1.0
0.8
0.6
0.4
0.2
0.0
0
1
2
3
4
Densidade da Exp(1)
2.3.1
Combinação de Exponenciais
y
2.0
y
2.0
1.5
1.5
1.0
1.0
0.5
0.5
0.0
0
1
2
3
4
0.0
5
x
0
1
f (x) = e−x
y
2
3
f (x) = 2e−2x
2.0
1.5
1.0
0.5
0.0
0
1
2
3
4
5
x
f (x) = 2e−x − 2e−2x
2.4
Exponencial Dupla ou de Laplace (α, β)
f (x) =
β −β|x−α|
, − ∞ < x < +∞
e
2
(−∞ < α < +∞, β > 0)
E[X] = α V [X] = 2/β 2 E[etX ] =
3
β 2 eαt
, |t| < β
β 2 − t2
4
5
x
y
0.5
0.4
0.3
0.2
0.1
-4
-3
-2
-1
0
1
2
3
4
Densidade da Laplace(0; 1)
2.5
F -Snedcor, F (m, n)
f (x) = B (m/2, n/2)
E[X] =
E[etX ]
:
−1
¡ m ¢m/2−1
m
nx
, x > 0 (m, n > 0)
¢
¡
(m+n)/2
n
1+ m
nx
n
2n2 (m + n − 2)
, n > 2; V [X] =
, n > 4;
n−2
m(n − 2)2 (n − 4)
não existe
Notas: X ∼ F (m, n) ⇒ X −1 ∼ F (n, m); F =
χ2 (m)/m
χ2 (n)/n
∼ F (m, n) (var. independentes)
y 0.6
0.5
0.4
0.3
0.2
0.1
0.0
0
1
2
3
4
Densidade da F (10; 4)
2.6
Gama (α, β)
β α α−1 −β x
e
, x > 0 (α, β > 0)
x
Γ(α)
Z ∞
Γ(α) =
β α xα−1 e−β x dx
f (x) =
0
Para α = n inteiro positivo: Γ(n) = (n − 1)! e F (x|n, β) = 1 −
E[X] =
α
α
V [X] = 2 E[etX ] =
β
β
Notas: Gama(1, β) ≡ Exp(β)
4
µ
β
β−t
¶α
E[X k ] =
n−1
X
k=0
k
(βx) e−βx
k!
Γ(α + k)
β α Γ(α)
f(x)
f(x)
x
x
Densidade da Gama(2; 2)
2.7
Densidade da Gama(3; 5)
Logística (α, β)
βe−β(x−α)
¡
¢2 , − ∞ < x < +∞
1 + e−β(x−α)
³
´−1
F (x) =
1 + e−β(x−α)
f (x) =
E[X] = α V [X] =
2.8
(−∞ < α < +∞, β > 0)
π2
E[etX ] = eαt Γ(1 − t/β)Γ(1 + t/β), |t| < β
3β 2
Gaussiana inversa (α, β)
½
¾
1
2
f (x) = α (2πβ)
x
exp −
(βx − α) , x > 0
2βx
r
½ µ
¶¾
2t
E[etX ] = exp α 1 − 1 −
β
−1/2
−3/2
(α, β)
y
Densidade da GI(α; β), α = β = 2−2 , 2−1 , 1, 2, 22 , 23 , 24 , 25
2.9
Lognormal (µ, σ 2 )
2
2
1
√ e−(log x−µ) /(2σ ) , − ∞ < x < +∞
xσ 2π
µ
¶
log x − µ
F (x) = Φ
σ
f (x|µ, σ 2 ) =
5
(−∞ < µ < +∞, σ > 0)
E[X] = eµ+σ
2
E[X k ] = ekµ+k
/2
2
V [X] = e2µ+σ
σ 2 /2
2
, k = 1, 2, ...
´
³ 2
E[etX ] : não existe
eσ − 1
Notas: Se X ∼LN(µ, σ 2 ) ⇔ Y = log X ∼N(µ, σ 2 )
y
Densidade da LN(0; 1)
2.10
Normal (µ, σ)
¡
¢−1/2 − 1 (x−µ)2
f (x) = 2πσ 2
e 2σ2
, −∞ < x < +∞ (−∞ < µ < +∞; σ > 0)
1
E[X] = µ; V [X] = σ 2 ; E[etX ] = eµt+ 2 σ
2 2
t
0.4
0.3
0.2
0.1
-3
-2
-1
0
1
2
3
x
Densidade da Normal(0; 1)
-3
-2
-1
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
1
2
3
4
-4
x
Densidades da Normal(0; 1) e Normal(1; 1)
-3
-2
-1
0
1
2
3
4
x
Densidades da Normal(0; 1) e Normal(0; 1.25)
6
2.11
Pareto (α, β)
αβ α
, x > 0 (α, β > 0)
(β + x)α+1
¶α
µ
β
,x>0
F (x) = 1 −
β+x
f (x) =
E[X] =
β
αβ 2
, α > 1; V [X] =
, α > 2; F.g.m. não existe
(α − 1)
(α − 1)2 (α − 2)
y
2.0
1.5
1.0
0.5
0.0
0
1
2
3
4
Densidade da Pareto(2; 1)
Notas:
• Se X ∼ Pareto(α, β) ⇒ Y = log (1 + X/β) ∼exp(α);
• Se X|θ ∼ exp(θ) e θ
2.12
Gama(α, β) ⇒ X
Pareto(α, β).
Pareto Generalizada (α, β, θ)
Γ (α + θ) αβ α xθ−1
, x > 0 (α, β > 0)
Γ (α) Γ (θ) (β + x)α+θ
µ
¶
x
F (x) = β θ, α;
,x>0
β+x
f (x) =
E[X k ] =
2.13
β k Γ (θ + k) Γ (α − k)
, −θ <k <α
Γ (α) Γ (θ)
E[etX ] : não existe
Qui-Quadrado χ2 (n)
³
´−1
f (x) = Γ(n/2)2n/2
xn/2−1 e−x/2 , x > 0 (n = 1, 2, ...)
E[X] = n V [X] = 2n E[etX ] =
Notas: χ2 (n) ≡ Gama(n/2, 1/2)
7
µ
1
1 − 2t
¶n/2
, t<
1
2
2.14
t-Student, t(n)
µ
¶−(n+1)/2
x2
, −∞ < x < +∞, (n = 1, 2, ...)
1+
n
Γ( n+1 )
f (x) = √ 2 n
nπΓ( 2 )
n
, n>2
n−2
Γ((k + 1) /2)Γ((n − k) /2)
1 · 3 · · · · · (k − 1)
E[X k ] = nk/2
= nk/2
, n > k, se k par
Γ(1/2)Γ(n/2)
(n − k) (n − k + 2) · · · (n − 2)
E[X] = 0, n > 1; V [X] =
E[X k ] = 0, se kímpar
E[etX ] : não existe
Notas: X e Y independentes, T =
N (0;1)
t
χ2(n) /n
∼ t(n); t(n) 2 ≡ F(1,n)
y 0.35
y
0.30
0.4
0.3
0.25
0.20
0.2
0.15
0.10
0.1
0.05
-3
-2
-1
0
1
2
3
-3
x
Densidades da t(1) [...] e t(3) [–]
2.15
-1
0
F (x) =
E[X] =
Notas: U(0; 1) ≡ Beta(1; 1)
1
2
Densidades da t(5) [–] e N (0; 1) [...]
Uniforme (a, b)
f (x) =
2.16
-2
1
, a < x < b (−∞ < a < b < ∞)
b−a
x−a
, a≤x≤b
b−a
a+b
ebt − eat
(b − a)2
V [X] =
E[etX ] =
2
12
(b − a)t
Weibull (α, β)
α
f (x) = αβxα−1 e−βx , x > 0 (α, β > 0)
α
F (x) = 1 − e−βx , x > 0
Γ (1 + 1/α)
E[X] =
β 1/α
Γ (1 + k/α)
E[X k ] =
β k/α
8
3
x