Providing Example that independence of events $A, B$ is dependent on Probability measure
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My example:
Let there be $3$ $6-$sided dice with each dice taking on either colour blue, green or red.
Let event $A:=$ the dice thrown is red.
and event $B:=$ the dice thrown shows $2$.
Let's then take $2$ probability measures $P, Q$. We set $P$ equal to uniform distribution.
It follows that $P(A cap B) = frac{1}{6 times 3}=frac{1}{18}$
and $P(A) times P(B)=frac{1}{3} * frac{1}{6} = frac{1}{18}$, so $A, B$ are naturally independent.
Now I am attempting to find a probability measure $Q$ such that:
$Q(A cap B) neq Q(A) times Q(B)$
My idea: perhaps $Q(C)=frac{2|Omega|-|C|}{|Omega|}$. But how can I be sure that it is $sigma-$additive, it is clearly "normed" as $Q(Omega)=1$.
$Q(A cap B)= frac{2 times 18-1}{18}>1$, which is not true for a probaility measure, right? Is my example wrong? Can you provide me with more intuitive examples?
probability probability-theory stochastic-processes independence
asked Nov 14 at 10:23
SABOY
495211
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up vote
0
down vote
favorite
My example:
Let there be $3$ $6-$sided dice with each dice taking on either colour blue, green or red.
Let event $A:=$ the dice thrown is red.
and event $B:=$ the dice thrown shows $2$.
Let's then take $2$ probability measures $P, Q$. We set $P$ equal to uniform distribution.
It follows that $P(A cap B) = frac{1}{6 times 3}=frac{1}{18}$
and $P(A) times P(B)=frac{1}{3} * frac{1}{6} = frac{1}{18}$, so $A, B$ are naturally independent.
Now I am attempting to find a probability measure $Q$ such that:
$Q(A cap B) neq Q(A) times Q(B)$
My idea: perhaps $Q(C)=frac{2|Omega|-|C|}{|Omega|}$. But how can I be sure that it is $sigma-$additive, it is clearly "normed" as $Q(Omega)=1$.
$Q(A cap B)= frac{2 times 18-1}{18}>1$, which is not true for a probaility measure, right? Is my example wrong? Can you provide me with more intuitive examples?
probability probability-theory stochastic-processes independence
asked Nov 14 at 10:23
SABOY
495211
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
My example:
Let there be $3$ $6-$sided dice with each dice taking on either colour blue, green or red.
Let event $A:=$ the dice thrown is red.
and event $B:=$ the dice thrown shows $2$.
Let's then take $2$ probability measures $P, Q$. We set $P$ equal to uniform distribution.
It follows that $P(A cap B) = frac{1}{6 times 3}=frac{1}{18}$
and $P(A) times P(B)=frac{1}{3} * frac{1}{6} = frac{1}{18}$, so $A, B$ are naturally independent.
Now I am attempting to find a probability measure $Q$ such that:
$Q(A cap B) neq Q(A) times Q(B)$
My idea: perhaps $Q(C)=frac{2|Omega|-|C|}{|Omega|}$. But how can I be sure that it is $sigma-$additive, it is clearly "normed" as $Q(Omega)=1$.
$Q(A cap B)= frac{2 times 18-1}{18}>1$, which is not true for a probaility measure, right? Is my example wrong? Can you provide me with more intuitive examples?
probability probability-theory stochastic-processes independence
asked Nov 14 at 10:23
SABOY
495211
My example:
Let there be $3$ $6-$sided dice with each dice taking on either colour blue, green or red.
Let event $A:=$ the dice thrown is red.
and event $B:=$ the dice thrown shows $2$.
Let's then take $2$ probability measures $P, Q$. We set $P$ equal to uniform distribution.
It follows that $P(A cap B) = frac{1}{6 times 3}=frac{1}{18}$
and $P(A) times P(B)=frac{1}{3} * frac{1}{6} = frac{1}{18}$, so $A, B$ are naturally independent.
Now I am attempting to find a probability measure $Q$ such that:
$Q(A cap B) neq Q(A) times Q(B)$
My idea: perhaps $Q(C)=frac{2|Omega|-|C|}{|Omega|}$. But how can I be sure that it is $sigma-$additive, it is clearly "normed" as $Q(Omega)=1$.
$Q(A cap B)= frac{2 times 18-1}{18}>1$, which is not true for a probaility measure, right? Is my example wrong? Can you provide me with more intuitive examples?
probability probability-theory stochastic-processes independence
probability probability-theory stochastic-processes independence
asked Nov 14 at 10:23
SABOY
495211
asked Nov 14 at 10:23
SABOY
495211
asked Nov 14 at 10:23
SABOY
495211
asked Nov 14 at 10:23
SABOY
495211
asked Nov 14 at 10:23
SABOY
495211
495211
add a comment |
add a comment |
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