Meaning of “Any two deterministic quantities are independent”
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I'm having trouble understanding this statement: "any two deterministic quantities are independent"
The example the text provides is as follows:
$$Prob(varnothingcapOmega) = Prob(varnothing) = 0 = Prob(varnothing)P(Omega) $$
Which proves $varnothing$ and $Omega$ are independent.
However, we cannot say that any event A and its complement, $A^C$, is necessarily independent(at least I don't think). My question is why are two deterministic quantities are independent? (Maybe I'm just not understanding what deterministic in this context means?)
Any guidance is greatly appreciated :)
probability probability-theory statistics independence
asked Dec 10 '18 at 16:47
Wilson GuoWilson Guo
312
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add a comment |
$begingroup$
I'm having trouble understanding this statement: "any two deterministic quantities are independent"
The example the text provides is as follows:
$$Prob(varnothingcapOmega) = Prob(varnothing) = 0 = Prob(varnothing)P(Omega) $$
Which proves $varnothing$ and $Omega$ are independent.
However, we cannot say that any event A and its complement, $A^C$, is necessarily independent(at least I don't think). My question is why are two deterministic quantities are independent? (Maybe I'm just not understanding what deterministic in this context means?)
Any guidance is greatly appreciated :)
probability probability-theory statistics independence
asked Dec 10 '18 at 16:47
Wilson GuoWilson Guo
312
$endgroup$
2
$begingroup$
I would guess that "deterministic" means the probability is either $0$ or $1$. (It's bound to happen or it will never happen.) In which case, two deterministic events are clearly independent.
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– saulspatz
Dec 10 '18 at 16:53
1
$begingroup$
@WilsonGuo Related to the saulspatz comment, can you prove the following? Let $A$ be an event with $P[A]=1$. Let $B$ be any other event. Use the definition of independent to prove that $A$ and $B$ are independent. [Hint: $B= (Bcap A) cup (B cap A^c)$. Or perhaps just prove $A^c$ and $B$ are independent.]
$endgroup$
– Michael
Dec 10 '18 at 17:02
add a comment |
$begingroup$
I'm having trouble understanding this statement: "any two deterministic quantities are independent"
The example the text provides is as follows:
$$Prob(varnothingcapOmega) = Prob(varnothing) = 0 = Prob(varnothing)P(Omega) $$
Which proves $varnothing$ and $Omega$ are independent.
However, we cannot say that any event A and its complement, $A^C$, is necessarily independent(at least I don't think). My question is why are two deterministic quantities are independent? (Maybe I'm just not understanding what deterministic in this context means?)
Any guidance is greatly appreciated :)
probability probability-theory statistics independence
asked Dec 10 '18 at 16:47
Wilson GuoWilson Guo
312
$endgroup$
I'm having trouble understanding this statement: "any two deterministic quantities are independent"
The example the text provides is as follows:
$$Prob(varnothingcapOmega) = Prob(varnothing) = 0 = Prob(varnothing)P(Omega) $$
Which proves $varnothing$ and $Omega$ are independent.
However, we cannot say that any event A and its complement, $A^C$, is necessarily independent(at least I don't think). My question is why are two deterministic quantities are independent? (Maybe I'm just not understanding what deterministic in this context means?)
Any guidance is greatly appreciated :)
probability probability-theory statistics independence
probability probability-theory statistics independence
asked Dec 10 '18 at 16:47
Wilson GuoWilson Guo
312
asked Dec 10 '18 at 16:47
Wilson GuoWilson Guo
312
asked Dec 10 '18 at 16:47
Wilson GuoWilson Guo
312
asked Dec 10 '18 at 16:47
Wilson GuoWilson Guo
312
asked Dec 10 '18 at 16:47
Wilson GuoWilson Guo
312
312
2
$begingroup$
I would guess that "deterministic" means the probability is either $0$ or $1$. (It's bound to happen or it will never happen.) In which case, two deterministic events are clearly independent.
$endgroup$
– saulspatz
Dec 10 '18 at 16:53
1
$begingroup$
@WilsonGuo Related to the saulspatz comment, can you prove the following? Let $A$ be an event with $P[A]=1$. Let $B$ be any other event. Use the definition of independent to prove that $A$ and $B$ are independent. [Hint: $B= (Bcap A) cup (B cap A^c)$. Or perhaps just prove $A^c$ and $B$ are independent.]
$endgroup$
– Michael
Dec 10 '18 at 17:02
add a comment |
2
$begingroup$
I would guess that "deterministic" means the probability is either $0$ or $1$. (It's bound to happen or it will never happen.) In which case, two deterministic events are clearly independent.
$endgroup$
– saulspatz
Dec 10 '18 at 16:53
1
$begingroup$
@WilsonGuo Related to the saulspatz comment, can you prove the following? Let $A$ be an event with $P[A]=1$. Let $B$ be any other event. Use the definition of independent to prove that $A$ and $B$ are independent. [Hint: $B= (Bcap A) cup (B cap A^c)$. Or perhaps just prove $A^c$ and $B$ are independent.]
$endgroup$
– Michael
Dec 10 '18 at 17:02
2
2
$begingroup$
I would guess that "deterministic" means the probability is either $0$ or $1$. (It's bound to happen or it will never happen.) In which case, two deterministic events are clearly independent.
$endgroup$
– saulspatz
Dec 10 '18 at 16:53
$begingroup$
I would guess that "deterministic" means the probability is either $0$ or $1$. (It's bound to happen or it will never happen.) In which case, two deterministic events are clearly independent.
$endgroup$
– saulspatz
Dec 10 '18 at 16:53
1
1
$begingroup$
@WilsonGuo Related to the saulspatz comment, can you prove the following? Let $A$ be an event with $P[A]=1$. Let $B$ be any other event. Use the definition of independent to prove that $A$ and $B$ are independent. [Hint: $B= (Bcap A) cup (B cap A^c)$. Or perhaps just prove $A^c$ and $B$ are independent.]
$endgroup$
– Michael
Dec 10 '18 at 17:02
$begingroup$
@WilsonGuo Related to the saulspatz comment, can you prove the following? Let $A$ be an event with $P[A]=1$. Let $B$ be any other event. Use the definition of independent to prove that $A$ and $B$ are independent. [Hint: $B= (Bcap A) cup (B cap A^c)$. Or perhaps just prove $A^c$ and $B$ are independent.]
$endgroup$
– Michael
Dec 10 '18 at 17:02
add a comment |
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$begingroup$
I would guess that "deterministic" means the probability is either $0$ or $1$. (It's bound to happen or it will never happen.) In which case, two deterministic events are clearly independent.
$endgroup$
– saulspatz
Dec 10 '18 at 16:53
1
$begingroup$
@WilsonGuo Related to the saulspatz comment, can you prove the following? Let $A$ be an event with $P[A]=1$. Let $B$ be any other event. Use the definition of independent to prove that $A$ and $B$ are independent. [Hint: $B= (Bcap A) cup (B cap A^c)$. Or perhaps just prove $A^c$ and $B$ are independent.]
$endgroup$
– Michael
Dec 10 '18 at 17:02