Distribuciones útiles (formato )

Unas distribuciones útiles
Distribuciones discretas
1. Binomial
X ∼ BI(n, θ) si
P (X = x) =
n
x
θx (1 − θ)n−x
x = 0, 1, . . . , n.
E[X] = nθ y V [X] = nθ(1 − θ).
2. Binomial Negativa
X ∼ BN (r, θ) si
P (X = x) =
x+r−1
x
θr (1 − θ)x
x = 0, 1, 2, . . .
(1−θ)
E[X] = r 1−θ
θ y V [X] = r θ 2 .
3. Geométrica
X ∼ GE(θ) si
P (X = x) = θ(1 − θ)x
E[X] =
1−θ
θ
y V [X] =
x = 0, 1, 2, . . .
1−θ
θ2 .
4. Hipergeométrica
X ∼ H(N, R, n) si
P (X = x) =
R
x
N −R
n−x
N
n
R
R
E[X] = n N
y V [X] = n N
1−
R
N
x = 0, 1, . . . , n
N −n
N −1 .
5. Poisson
X ∼ P(θ) si
P (X = x) =
θx e−θ
x!
E[X] = θ = V [X].
1
x = 0, 1, 2, . . .
6. Uniforme Discreta
X ∼ UD[θ1 , θ2 ] si
1
θ2 − θ1 + 1
P (X = x) =
E[X] =
θ1 +θ2
2
y V [X] =
x = θ1 , θ1 + 1, . . . , θ2 .
(θ2 −θ1 )(θ2 −θ1 +2)
.
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Distribuciones continuas
1. Behrens Fisher
X ∼ BF(ν1 , ν2 , θ) si X = T1 cos θ − T2 sin θ y Ti ∼ T (νi , 0, 1).
E[X] = 0 y V [X] =
ν1 sin2 θ
ν1 −2
+
ν2 cos2 θ
ν2 −2
para ν1 , ν2 > 2.
2. Beta
X| ∼ B(α, β) si
f (x) =
donde B(α, β) =
E[X] =
α
α+β
1
xα−1 (1 − x)β−1
B(α, β)
0 < x < 1,
Γ(α)Γ(β)
Γ(α+β) .
y V [X] =
αβ
(α+β)2 (α+β+1) .
3. Cauchy
X ∼ C(µ, σ 2 ) si
f (x) =
1
σ
π σ 2 + (x − µ)2
∀ x.
No existen los momentos.
4. Chi-cuadrada
X ∼ χ2ν si X ∼ G(ν/2, 1/2).
2−ν/2 ν/2−1 −x/2
x
e
Γ(ν/2)
f (x) =
x > 0.
E[X] = ν y V [X] = 2ν.
5. Chi-cuadrada invertida
X ∼ Iχ2ν si 1/X ∼ χ2ν . E[X] =
ν > 4.
1
ν−2
si ν > 2 y V [X] =
6. Exponencial
X ∼ E(θ) si
f (x) = θe−θx
E[X] =
1
θ
y V [X] =
1
θ2 .
2
x > 0.
2
(ν−2)2 (ν−4)
si
7. F de Fisher
X ∼ F(α, β) si
1
f (x) =
B(α/2, β/2)
E[X] =
α
β−2
α/2
− α+β
2
α
α
α/2−1
x
1+ x
β
β
si β > 2 y V [X] =
2β 2 (α+β−2)
α(β−2)2 (β−4)
∀ x.
si β > 4.
8. Gamma
X| ∼ G(α, β) si
f (x) =
E[X] =
α
β
y V [X] =
β
(βx)α−1 e−βx
Γ(α)
x > 0.
α
β2 .
9. Gamma Invertida
X ∼ GI(α, β) si X −1 ∼ G(α, β).
f (x) =
E[X] =
β
α−1
β α −(α+1) −β/x
x
e
Γ(α)
si α > 1 y V [X] =
β2
(α−1)2 (α−2)
x > 0.
si α > 2.
10. Normal
X ∼ N (µ, σ 2 ) si
f (x) =
2
1
1
√ e− 2σ2 (x−µ)
σ 2π
∀x.
E[X] = µ y V [X] = σ 2 .
11. Pareto
X ∼ PA(α, β) si
f (x) = βαβ x−β−1
E[X] =
αβ
β−1
si β > 1 y V [X] =
2
α β
(β−1)(β−2)
x > α.
si β > 2.
12. T de Student
X ∼ T (ν, µ, σ 2 ) si
Γ((ν + 1)/2)
√
f (x) =
Γ(ν/2) νπσ
E[X] = µ si ν > 1 y V [X] =
1
1+
ν
ν
2
ν−2 σ
x−µ
σ
2 !−(ν+1)/2
∀ x.
si ν > 2.
La distribución T (ν, 0, 1) es la distribucin t de Student estandar.
3
13. Uniforme
X ∼ U(0, θ) si
f (x) =
E[X] =
θ
2
1
θ
0 < x < θ.
θ2
12 .
y V [X] =
Distribuciones multivariables
1. Dirichlet
X = (X1 , . . . , Xp )T ∼ D(θ) si
Pp
p
Γ ( i=1 θi ) Y θ1 −1
x
f (x) =
Γ(θ1 ) · · · Γ(θp ) i=1 i
θ
E[Xj ] = Pp j
i=1
θi
y V [Xj ] = (Pp
i=1
para 0 < x1 < 1,
θj (1−θ
Pjp)
θi ) 2 (
i=1
θi +1)
Pp
i=1
xi = 1
.
2. Normal multivariante
X = (X1 , . . . , Xp )T ∼ N (µ, Σ) si
f (x) =
1
(2π)p/2 |Σ|1/2
1
T −1
exp − (x − µ) Σ (x − µ)
2
E[X] = µ y V [X] = Σ.
3. Multinomial
X = (X1 , . . . , Xp )T ∼ MN (n, θ) si
n!
p
Y
i=1
xi ! i=1
P (X = x) = Qp
θixi
donde
Pp
i=1
xi = n y 0 < θi < 1 ∀i
4. Wishart
V ∼ W(ν, Σ) si
f (V) =
2
νk/2 k(k−1)/4
π
!−1
k
Y
ν+1−i
1
Γ
|V |−ν/2 |Σ|(ν−k−1)/2 exp − tr(Σ−1 V)
2
2
i=1
donde k = dimV.
E[V] = νΣ.
5. Wishart invertida
V ∼ WI(ν, Σ−1 ) si V ∼ W(ν, Σ−1 ).
E[V] = (ν − k − 1)−1 Σ.
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