Natural filtration and coin tossing
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I have a problem understanding a probably easy task from a book I'm reading at the moment:
"Show, that the natural filtration $ F_2 $, generated by observing the events $ A_1 $ = {(H,H),(H,T)} and $ A_2 $ ={(H,T)} has only eight elements."
In the book it is mentioned, that a $ sigma $ - algebra generated by an event A is F = {$ emptyset,A,A^c, Omega $ }.
If I apply this to $ A_1 $ I think I get the following result:
$ sigma(A_1) =$ {$ emptyset, {HH,HT}, {HH, TT}, {TT, HT}, {HH, TH},{HT, TH},{TH, TT}, Omega}$,
this solution would have eight elements, but what is with $ A_2 $?
I think I have a problem understanding the concept of (natural) filtration, I know that each filtration "level" contains the information of the previous ones but how for example is the set $ {HT, TH, TT} in F_2 $? Does this set describes the situation that after toss number two we know that at least one toss was tails?
How can I imagine the sets $ sigma (A_1) $ and $ sigma(A_2) $? Which situation do they describe?
Thank you in advance!
probability filtrations
asked Jan 17 at 18:02
IVSTICEIVSTICE
11
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add a comment |
$begingroup$
I have a problem understanding a probably easy task from a book I'm reading at the moment:
"Show, that the natural filtration $ F_2 $, generated by observing the events $ A_1 $ = {(H,H),(H,T)} and $ A_2 $ ={(H,T)} has only eight elements."
In the book it is mentioned, that a $ sigma $ - algebra generated by an event A is F = {$ emptyset,A,A^c, Omega $ }.
If I apply this to $ A_1 $ I think I get the following result:
$ sigma(A_1) =$ {$ emptyset, {HH,HT}, {HH, TT}, {TT, HT}, {HH, TH},{HT, TH},{TH, TT}, Omega}$,
this solution would have eight elements, but what is with $ A_2 $?
I think I have a problem understanding the concept of (natural) filtration, I know that each filtration "level" contains the information of the previous ones but how for example is the set $ {HT, TH, TT} in F_2 $? Does this set describes the situation that after toss number two we know that at least one toss was tails?
How can I imagine the sets $ sigma (A_1) $ and $ sigma(A_2) $? Which situation do they describe?
Thank you in advance!
probability filtrations
asked Jan 17 at 18:02
IVSTICEIVSTICE
11
$endgroup$
$begingroup$
Please can somebody help me here?
$endgroup$
– IVSTICE
Jan 24 at 10:56
add a comment |
$begingroup$
I have a problem understanding a probably easy task from a book I'm reading at the moment:
"Show, that the natural filtration $ F_2 $, generated by observing the events $ A_1 $ = {(H,H),(H,T)} and $ A_2 $ ={(H,T)} has only eight elements."
In the book it is mentioned, that a $ sigma $ - algebra generated by an event A is F = {$ emptyset,A,A^c, Omega $ }.
If I apply this to $ A_1 $ I think I get the following result:
$ sigma(A_1) =$ {$ emptyset, {HH,HT}, {HH, TT}, {TT, HT}, {HH, TH},{HT, TH},{TH, TT}, Omega}$,
this solution would have eight elements, but what is with $ A_2 $?
I think I have a problem understanding the concept of (natural) filtration, I know that each filtration "level" contains the information of the previous ones but how for example is the set $ {HT, TH, TT} in F_2 $? Does this set describes the situation that after toss number two we know that at least one toss was tails?
How can I imagine the sets $ sigma (A_1) $ and $ sigma(A_2) $? Which situation do they describe?
Thank you in advance!
probability filtrations
asked Jan 17 at 18:02
IVSTICEIVSTICE
11
$endgroup$
I have a problem understanding a probably easy task from a book I'm reading at the moment:
"Show, that the natural filtration $ F_2 $, generated by observing the events $ A_1 $ = {(H,H),(H,T)} and $ A_2 $ ={(H,T)} has only eight elements."
In the book it is mentioned, that a $ sigma $ - algebra generated by an event A is F = {$ emptyset,A,A^c, Omega $ }.
If I apply this to $ A_1 $ I think I get the following result:
$ sigma(A_1) =$ {$ emptyset, {HH,HT}, {HH, TT}, {TT, HT}, {HH, TH},{HT, TH},{TH, TT}, Omega}$,
this solution would have eight elements, but what is with $ A_2 $?
I think I have a problem understanding the concept of (natural) filtration, I know that each filtration "level" contains the information of the previous ones but how for example is the set $ {HT, TH, TT} in F_2 $? Does this set describes the situation that after toss number two we know that at least one toss was tails?
How can I imagine the sets $ sigma (A_1) $ and $ sigma(A_2) $? Which situation do they describe?
Thank you in advance!
probability filtrations
probability filtrations
asked Jan 17 at 18:02
IVSTICEIVSTICE
11
asked Jan 17 at 18:02
IVSTICEIVSTICE
11
asked Jan 17 at 18:02
IVSTICEIVSTICE
11
asked Jan 17 at 18:02
IVSTICEIVSTICE
11
asked Jan 17 at 18:02
IVSTICEIVSTICE
11
11
$begingroup$
Please can somebody help me here?
$endgroup$
– IVSTICE
Jan 24 at 10:56
add a comment |
$begingroup$
Please can somebody help me here?
$endgroup$
– IVSTICE
Jan 24 at 10:56
$begingroup$
Please can somebody help me here?
$endgroup$
– IVSTICE
Jan 24 at 10:56
$begingroup$
Please can somebody help me here?
$endgroup$
– IVSTICE
Jan 24 at 10:56
add a comment |
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– IVSTICE
Jan 24 at 10:56