Matematica Actuarial No vida
Páginas: 3 (529 palabras)
Publicado: 3 de diciembre de 2013
Distribuciones discretas
POISSON
Po(λ)
λ
p0 = e
k
k!
P[ N = k ]
POISSON-INVERSA
GAUSSIANA
PIG (µ,ß)
⋅ e −λ
λ > 0 k = 0,1,2,...
BINOMIAL NEGATIVA
BN(a,b)
1
µ
⋅ 1−(1+ 2β ) 2
β
p1 = µ ⋅ p0 ⋅ (1 + 2 β )
−
Γ(a + k ) 1 b
⋅
⋅
Γ ( a ) ⋅ k! 1 + b 1 + b
a,b > 0 k = 0,1,2,...
a
1
2
(1 + 2 β ) ⋅ k ⋅ ( k − 1) ⋅ pk=
= β ⋅ ( k − 1) ⋅ ( 2k − 3) ⋅ pk −1 +
+ µ 2 ⋅ pk − 2
k=2,3,…
[ ]
P( z ) = E z N
e λ ⋅( z −1)
M N ( z ) = E e
z⋅N
e
λ ⋅( e z −1)
e
(
µ
⋅ 1− 1+ 2β (1− z )
β(
1
1 − b ⋅ ( z − 1)
1+ b
| z |<
b
)
)
µ
⋅ 1− 1+ 2β 1− e z
eβ
1
1 − b ⋅ (e z − 1)
E [N ]
λ
µ
a ⋅b
V [N ]λ
µ ⋅ (1 + β )
a ⋅ b ⋅ (1 + b)
Estimadores parámetros distribuciones discretas (método momentos)
POISSON
Po(λ)
ˆ
λ=N
ˆ
a=
BINOMIAL NEGATIVA
BN(a,b)
POISSON-INVERSAGAUSSIANA
PIG (µ,ß)
a
N2
2
SN − N
2
ˆ = SN − N
b
N
2
ˆ S
ˆ
µ=N = N
β
N
−1
a
k
Distribuciones continuas
EXPONENCIAL
Ex(b)
b ⋅ e − b⋅ x
f (x)
1
σ 2πx>0
b>0
F ( x)
b
b−z
k!
bk
1
b
1
b2
αk
E[X ]
V [X ]
F ( x)
1
1 x−µ 2
−
2 σ
⋅e
σx 2π
1 ln x − µ 2
−
2 σ
x>0 σ>0
-∞ < μ < ∞ ln x − µ
Φ
x>0
σ
FX ( x ) = Φ ( x )
e
µ ⋅z +
σ 2z2
2
e
µ
µk +
e
σ 2k 2
2
µ+
σ2
2
e 2 µ +σ (e σ − 1)
2
σ
GAMMA
Ga(a,b)
f (x)
⋅eLOGNORMAL
LN(μ,σ)
X ~N(0,1)
1 − e − b⋅ x
M X (z )
NORMAL
N(μ,σ)
ba
⋅ x a −1 ⋅ e − bx
Γ( a )
x > 0 a, b > 0
Ga( x; a, b)
2
2
INVERSA GAUSSIANA
IG(μ,β)
(
µ 2πβx 3
)−
1
2
⋅ e−
(x − µ )2
2βx
x>0
β,μ>0
1
−
µβ −1
−( x + µ )
2 ⋅ ( x − µ ) + +e2
Φ ( β x )
⋅Φ
1
( β x )2
x>0
a
M X (z )...
POISSON
Po(λ)
λ
p0 = e
k
k!
P[ N = k ]
POISSON-INVERSA
GAUSSIANA
PIG (µ,ß)
⋅ e −λ
λ > 0 k = 0,1,2,...
BINOMIAL NEGATIVA
BN(a,b)
1
µ
⋅ 1−(1+ 2β ) 2
β
p1 = µ ⋅ p0 ⋅ (1 + 2 β )
−
Γ(a + k ) 1 b
⋅
⋅
Γ ( a ) ⋅ k! 1 + b 1 + b
a,b > 0 k = 0,1,2,...
a
1
2
(1 + 2 β ) ⋅ k ⋅ ( k − 1) ⋅ pk=
= β ⋅ ( k − 1) ⋅ ( 2k − 3) ⋅ pk −1 +
+ µ 2 ⋅ pk − 2
k=2,3,…
[ ]
P( z ) = E z N
e λ ⋅( z −1)
M N ( z ) = E e
z⋅N
e
λ ⋅( e z −1)
e
(
µ
⋅ 1− 1+ 2β (1− z )
β(
1
1 − b ⋅ ( z − 1)
1+ b
| z |<
b
)
)
µ
⋅ 1− 1+ 2β 1− e z
eβ
1
1 − b ⋅ (e z − 1)
E [N ]
λ
µ
a ⋅b
V [N ]λ
µ ⋅ (1 + β )
a ⋅ b ⋅ (1 + b)
Estimadores parámetros distribuciones discretas (método momentos)
POISSON
Po(λ)
ˆ
λ=N
ˆ
a=
BINOMIAL NEGATIVA
BN(a,b)
POISSON-INVERSAGAUSSIANA
PIG (µ,ß)
a
N2
2
SN − N
2
ˆ = SN − N
b
N
2
ˆ S
ˆ
µ=N = N
β
N
−1
a
k
Distribuciones continuas
EXPONENCIAL
Ex(b)
b ⋅ e − b⋅ x
f (x)
1
σ 2πx>0
b>0
F ( x)
b
b−z
k!
bk
1
b
1
b2
αk
E[X ]
V [X ]
F ( x)
1
1 x−µ 2
−
2 σ
⋅e
σx 2π
1 ln x − µ 2
−
2 σ
x>0 σ>0
-∞ < μ < ∞ ln x − µ
Φ
x>0
σ
FX ( x ) = Φ ( x )
e
µ ⋅z +
σ 2z2
2
e
µ
µk +
e
σ 2k 2
2
µ+
σ2
2
e 2 µ +σ (e σ − 1)
2
σ
GAMMA
Ga(a,b)
f (x)
⋅eLOGNORMAL
LN(μ,σ)
X ~N(0,1)
1 − e − b⋅ x
M X (z )
NORMAL
N(μ,σ)
ba
⋅ x a −1 ⋅ e − bx
Γ( a )
x > 0 a, b > 0
Ga( x; a, b)
2
2
INVERSA GAUSSIANA
IG(μ,β)
(
µ 2πβx 3
)−
1
2
⋅ e−
(x − µ )2
2βx
x>0
β,μ>0
1
−
µβ −1
−( x + µ )
2 ⋅ ( x − µ ) + +e2
Φ ( β x )
⋅Φ
1
( β x )2
x>0
a
M X (z )...
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