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ESTADISTICA
PIEGI / UTFSMCS
20 de Junio 2002
TAREA ESTADISTICA
DEMOSTRACIONES DE PROBABILIDAD
Tarea:
Si A y B son independientes:
A-B y B-A son independientes?
A y B-A son independientes?
A y A-B son independientes?
B y B-a son independientes?
B y A-B son independientes?
Desarrollo:
Se sabe que si A y B son independientes: A y Bc son independientes
Ac y B son independientes
Ac y Bc son independientes
A-B y B-A son independientes:
Sea T= A-B
W=B-A
Sean T y W independientes
P(T
W) = P[(A-B)
(B-A)]
P(T
W) = P[(A
Bc)
(Ac
B)]
P(T
W) = P[(A
Ac)
(B
Bc)]
P(T
W) = P[(
)
(
)]
P(T
W) = 0
P(T)*P(W) = 0 ecc.(1)
Por otro lado:
P(T) = P(A-B)
P(T) = P(A
Bc)
P(T) = P(A) * P(Bc)
P(W) = P(B-A)
P(W) = P(B
Ac)
P(W) = P(B) * P(Ac)
En consecuencia:
P(T)*P(W) = P(A) * P(Bc) * P(B) * P(Ac)
P(T)*P(W) = P(A) * (1 - P(B)) * P(B) * (1 - P(A))
Ordenando:
P(T)*P(W) = P(A) * P(B) * (1 - P(B)) * (1 - P(A)) ecc.(2)
No son independientes ya que ecc.(1) y ecc.(2) no son iguales.
2) ¿A y (B-A) son independientes?
Sea T= A
W=B-A
Sean T y W independientes
(T
W) = A
(B-A)]
(T
W) = A
(B
Ac)
(T
W) = A
Ac
B
P(T
W) = P(
B )
P(T
W) = P(
)
P(T
W) = 0
P(T)*P(W) = 0 ecc.(3)
Por otro lado:
P(T) = P(A)
P(W) = P(B
Ac)
P(T)*P(W) = P(A) * P(B
Ac)
P(T)*P(W) = P(A) * P(B) * P(Ac)
P(T)*P(W) = P(A) * P(B) * [1-P(A)]
P(T)*P(W) = P(A) * P(B) - P(B) * [P(A)]2 ecc.(4)
Como ecc(3) no es igual a ecc(4) no son independientes
3)¿A y (A-B) son independientes?
Sea T= A
W=A-B
Sean T y W independientes
(T
W) = A
(A-B)]
(T
W) = A
(Bc
A)
P(T
W) = P(A
A
Bc)
P(T
W) = P(A
Bc)
P(T)*P(W) = P(A) * P(Bc) ecc.(5)
Por otro lado:
P(T) = P(A)
P(W) = P(A
Bc)
P(W) = P(A) * P(Bc)
P(T)*P(W) = P(A) * P(A) * P(Bc)
P(T)*P(W) = [P(A)]2 * P(Bc) ecc.(6)
Como ecc(5) no es igual a ecc(6) no son independientes, excepto para P(A) = 1
4)¿B y (B-A) son independientes?
Sea T= B
W=B - A
Sean T y W independientes
(T
W) = B
(B-A)]
(T
W) = B
(B
Ac)
(T
W) = (B
B
Ac )
(T
W) = (B
Ac)
P(T
W) = P(B
Ac)
P(T)*P(W) = P(B) * P(Ac) ecc.(7)
Por otro lado:
P(T) = P(B)
P(W) = P(B - A)
P(W) = P(B
Ac)
P(W) = P(B) * P(Ac)
P(T)*P(W) = P(B) * P(B) * P(Ac)
P(T)*P(W) = [P(B)]2 * P(Ac) ecc.(8)
Como ecc(7) no es igual a ecc(8) no son independientes, excepto para P(B) = 1
5)¿B y (A-B) son independientes?
Sea T= B
W= A - B
Sean T y W independientes
(T
W) = B
(A-B)]
(T
W) = B
(A
Bc)
(T
W) = (B
A
Bc )
P(T
W) = P(B
Bc
A)
P(T
W) = P(
A)
P(T)*P(W) = P(
)
P(T)*P(W) = 0 ecc.(9)
Por otro lado:
P(T) = P(B)
P(W) = P(A - B)
P(W) = P(A
Bc)
P(W) = P(A) * P(Bc)
P(T)*P(W) = P(B) * P(A) * P(Bc)
P(T)*P(W) = P(B) * P(A) * [1 - P(B)]
P(T)*P(W) = P(B) * P(A) - P(A)* [P(B)]2 ecc.(10)
Como ecc(9) no es igual a ecc(10) no son independientes.
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Enviado por: | El remitente no desea revelar su nombre |
Idioma: | castellano |
País: | Chile |