05 Formulario

Universidad T´ecnica Federico Santa Mar´ıa
Departamento de Matem´atica
Casa Central

Esteban Henr´ıquez C.
Vectores Aleatorios
2do. Semestre 2010

MAT 042 – PROBABILIDAD Y ESTAD´ISTICA INDUSTRIALFormulario Vectores Aleatorios
X = (X, Y) es absolutamente continuo

X = (X, Y) es discreto

fX,Y (x, y) ≥ 0

fX,Y (x, y) ≥ 0
∞

∞

∞

∞

−∞

−∞

fX,Y (x, y) = 1

fX,Y (x, y) dy dx = 1

y=−∞ x=−∞

P[X =x, Y = y] = fX,Y (x, y)

P[X = x, Y = y] = 0

Marginal de X

Marginal de Y

F X (x) = l´ım F X,Y (x, y)
y→∞

∞





fX,Y (x, y) , Discreto


 y=−∞
fX (x) = 

∞





fX,Y (x, y) dy ,Continuo

−∞
 ∞ ∞





xr fX,Y (x, y) , Disc.



x=−∞ y=−∞
E [X r ] = 





xr fX,Y (x, y) dydx , Cont.


2

FY (y) = l´ım F X,Y (x, y)
x→∞

∞





fX,Y (x, y) , Discreto


x=−∞
fY (y) = 

∞





fX,Y (x, y) dx , Continuo

 −∞
∞
∞





yr fX,Y (x, y) , Disc.



x=−∞ y=−∞
E [Y r ] = 





yr fX,Y (x, y) dydx , Cont.


2

V [X] = E[X ] − (E [X])

V[Y] = E[Y 2 ] − (E [Y])2

R

2

2

E g(X, Y)

R

Sea g : R2 → R, entonces:
 ∞ ∞





g(x, y) fX,Y (x, y) , Disc.


 x=−∞ y=−∞
= 





g(x, y) fX,Y (x, y) dydx , Cont.


R2

Cov(X, Y)= E[XY] − E[X]E[Y]
Cov(X, X) = V[X]
Condicional: Y/X = x
fX,Y (x, y)
f (x)
 X ∞





yr fY/X=x (y) , Disc.


 y=−∞
r
E [Y /X = x] = 

∞





yr fY/X=x (y) dy , Cont.

fY/X=x (y) =



 fX,Y (x, y) = fX (x) fY (y)
X⊥Y ⇔ 

 F X,Y (x, y) = F X (x)FY (y)
X⊥Y ⇒ Cov(X, Y) = 0
ρX,Y
ρX,Y

Cov(X, Y)
√
V[X]V[Y]
≤ 1
=

E[aX ± bY] = aE[X] ± bE[Y]
V[aX ± bY] = a2 V[X] + b2 V[Y]
±2abCov(X,Y)
E[Y] = E[E [Y/X = x]]
E[XY/X = x] = X E [Y/X = x]

−∞

V [Y/X = x] = E[Y 2 /X = x] − (E [Y/X = x])2
An´alogo para el caso X/Y = y

V[Y] = VEntre + VDentro
= V[E [Y/X = x]] +
+E[V [Y/X = x]]Cov(aV1 + bV2 , cW1 + dW2 ) = acCov(V1 , W1 ) + adCov(V1 , W2 ) + bcCov(V2 , W1 ) + bdCov(V2 , W2 )
LATEX 2ε \ EHC – 02 de noviembre de 2010

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