Taller Empresarial

Universidad T´ cnica Federico Santa Mar´a
e
ı
´
Departamento de Matematica
Casa Central

Esteban Henr´quez Castro
ı
Modelos de Probabilidad
Primer Semestre 2012

MAT 031 – ESTAD´STICA
IFormulario Modelos de Probabilidad
X ∼ Hip(N, M, n)

X ∼ Bin(n, p)

X ∼ Ber( p)
Rec(X ) = {0, 1}
fX ( x) = p x (1 − p)1− x
E[ X ] = p
V[X ] = p(1 − p)

Rec(X ) = {0, . . . , m´n(n, M )}
ıRec(X ) = {0, 1, . . . , n}
nx
p (1 − p)n− x
f X ( x) =
x
E[X ] = np

f X ( x) =

fX ( x) = p (1 − p) x−1
E[ X ] =
V[ X ] =

1
p
1− p
p2

X (t) ∼ Poisson(θt)

Rec(X ) = RΓ(r) =

−θ x

u=0

e−θt (θt)u
u!

∞
0

Gamma(r, θ)

Rec(X )

=

f X ( x)

=

E[ X ]

=

R+
θ r r −1
x exp(−θ x)
Γ(r)

xr−1 e− x dx

Γ(r) = (r − 1) Γ(r − 1)
√
π
Γ1 =
2F X ( x) = 1 − e−θ x
X ∼ Unif(a, b)
Rec(X ) = (a, b)
1
f X ( x) =
b−a
E[X ] = 1 (a + b)
2
V[ X ] =

F X ( x) =
∼

Γ(r) = (r − 1)!

1
θ
1
θ2

1
(b
12

x

F X ( x) = 1 −P[S x ≤ r − 1]
S x ∼ Bin( x, p)

Funci´ n Gamma:
o

+

V[ X ] =

= θt

X

X ∼ exp(θ)

E[ X ] =

= {0, 1, . . .}
e−θt (θt) x
=
x!
= θt

Rec(X (t))
Rec(X ) = {r, r + 1, . . .}x−1 r
f X ( x)
f X ( x) =
p (1 − p) x−r
r−1
r
E[X (t)]
E[ X ] = p
V[X (t)]
V[X ] = r (1− p)
2
p

F X ( x) = 1 − (1 − p) x

f X ( x) = θ e

M
N
n M N − M N −n
N N N −1

V[ X ] =

X∼ BN(r, p)
Rec(X ) = {1, 2, . . .}

N
n

E[ X ] = n

V[X ] = np(1 − p)

X ∼ Geo( p)

N−M
n− x

M
x

=

V[ X ]

r ∈N

=

F X ( x)

X ∼ Weibull(α, λ)

Rec(X ) = R
−(λ x)αfX ( x) = αλα xα−1 e
E[ X ] =

− a)
x−a
F X ( x) =
b−a

X ∼ N (µ, σ2 )
X−µ
Z=
∼ N (0, 1)
σ
FZ (z) = Φ (z)
Φ(a) = 1 − Φ(−a)

1 − P[N (t) ≤ r − 1]
N (t) ∼ Poisson(θt)
X ∼ Normal(µ, σ2)

Rec(X ) = R+

2

r
θ
r
θ2

1
Γ
λ

1+

V[ X ] =

1
λ2

Γ 1+

f X ( x) =

1
α
2
α

− Γ 1+

1
α

E[ X ] = µ

2

F X ( x) = Φ

α

fT (t) = −R (t)
fT...