Is collection of subsets of $Omega$ that are determined by first n number of coin tosses a $sigma$-algebra?
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Let $F_n$ be the collection of subsets of $Omega$ whose occurrence can be decided by looking at the first n tosses. How can I show $F_n$ is a $sigma$- algebra?
probability-theory measure-theory
edited Aug 26 at 23:58
spaceisdarkgreen
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asked Aug 26 at 23:52
D.S
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Let $F_n$ be the collection of subsets of $Omega$ whose occurrence can be decided by looking at the first n tosses. How can I show $F_n$ is a $sigma$- algebra?
probability-theory measure-theory
edited Aug 26 at 23:58
spaceisdarkgreen
28.6k21548
asked Aug 26 at 23:52
D.S
84
I don't understand what "determined by the first n tosses" means.
– DanielWainfleet
Aug 26 at 23:54
For example,Let $A_1$ be the set of all elements of Ω such that there are exactly 2 heads during the first 4 coin tosses. Then, $A_1 ∈ F_4$.
– D.S
Aug 26 at 23:56
1
This problem seems to require you to do two things. First, say exactly what $F_n$ is --- not "for example" as in your comment but a mathematically precise definition. Second, combine that and the definition of "$sigma$-algebra" to determine exactly what you need to prove. Technically, there's a third step, namely to prove those things, but I think that, once you've got the goals written down exactly, they will be very easy to prove. (P.S. I'm not saying that your "For example" comment won't clarify for @DanielWainfleet what you intended, but something more precise is needed for a proof.)
– Andreas Blass
Aug 27 at 0:29
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $F_n$ be the collection of subsets of $Omega$ whose occurrence can be decided by looking at the first n tosses. How can I show $F_n$ is a $sigma$- algebra?
probability-theory measure-theory
edited Aug 26 at 23:58
spaceisdarkgreen
28.6k21548
asked Aug 26 at 23:52
D.S
84
Let $F_n$ be the collection of subsets of $Omega$ whose occurrence can be decided by looking at the first n tosses. How can I show $F_n$ is a $sigma$- algebra?
probability-theory measure-theory
edited Aug 26 at 23:58
spaceisdarkgreen
28.6k21548
asked Aug 26 at 23:52
D.S
84
edited Aug 26 at 23:58
spaceisdarkgreen
28.6k21548
edited Aug 26 at 23:58
spaceisdarkgreen
28.6k21548
edited Aug 26 at 23:58
spaceisdarkgreen
28.6k21548
28.6k21548
asked Aug 26 at 23:52
D.S
84
asked Aug 26 at 23:52
D.S
84
asked Aug 26 at 23:52
D.S
84
84
I don't understand what "determined by the first n tosses" means.
– DanielWainfleet
Aug 26 at 23:54
For example,Let $A_1$ be the set of all elements of Ω such that there are exactly 2 heads during the first 4 coin tosses. Then, $A_1 ∈ F_4$.
– D.S
Aug 26 at 23:56
1
This problem seems to require you to do two things. First, say exactly what $F_n$ is --- not "for example" as in your comment but a mathematically precise definition. Second, combine that and the definition of "$sigma$-algebra" to determine exactly what you need to prove. Technically, there's a third step, namely to prove those things, but I think that, once you've got the goals written down exactly, they will be very easy to prove. (P.S. I'm not saying that your "For example" comment won't clarify for @DanielWainfleet what you intended, but something more precise is needed for a proof.)
– Andreas Blass
Aug 27 at 0:29
add a comment |Â
I don't understand what "determined by the first n tosses" means.
– DanielWainfleet
Aug 26 at 23:54
For example,Let $A_1$ be the set of all elements of Ω such that there are exactly 2 heads during the first 4 coin tosses. Then, $A_1 ∈ F_4$.
– D.S
Aug 26 at 23:56
1
This problem seems to require you to do two things. First, say exactly what $F_n$ is --- not "for example" as in your comment but a mathematically precise definition. Second, combine that and the definition of "$sigma$-algebra" to determine exactly what you need to prove. Technically, there's a third step, namely to prove those things, but I think that, once you've got the goals written down exactly, they will be very easy to prove. (P.S. I'm not saying that your "For example" comment won't clarify for @DanielWainfleet what you intended, but something more precise is needed for a proof.)
– Andreas Blass
Aug 27 at 0:29
I don't understand what "determined by the first n tosses" means.
– DanielWainfleet
Aug 26 at 23:54
I don't understand what "determined by the first n tosses" means.
– DanielWainfleet
Aug 26 at 23:54
For example,Let $A_1$ be the set of all elements of Ω such that there are exactly 2 heads during the first 4 coin tosses. Then, $A_1 ∈ F_4$.
– D.S
Aug 26 at 23:56
For example,Let $A_1$ be the set of all elements of Ω such that there are exactly 2 heads during the first 4 coin tosses. Then, $A_1 ∈ F_4$.
– D.S
Aug 26 at 23:56
1
1
This problem seems to require you to do two things. First, say exactly what $F_n$ is --- not "for example" as in your comment but a mathematically precise definition. Second, combine that and the definition of "$sigma$-algebra" to determine exactly what you need to prove. Technically, there's a third step, namely to prove those things, but I think that, once you've got the goals written down exactly, they will be very easy to prove. (P.S. I'm not saying that your "For example" comment won't clarify for @DanielWainfleet what you intended, but something more precise is needed for a proof.)
– Andreas Blass
Aug 27 at 0:29
This problem seems to require you to do two things. First, say exactly what $F_n$ is --- not "for example" as in your comment but a mathematically precise definition. Second, combine that and the definition of "$sigma$-algebra" to determine exactly what you need to prove. Technically, there's a third step, namely to prove those things, but I think that, once you've got the goals written down exactly, they will be very easy to prove. (P.S. I'm not saying that your "For example" comment won't clarify for @DanielWainfleet what you intended, but something more precise is needed for a proof.)
– Andreas Blass
Aug 27 at 0:29
add a comment |Â
1 Answer
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So, $Omega$ is the infinite product $Pi_iin mathbbN H, T$. The sets you're looking at are such that if $x in A in F_n$ then $y in A$ if $x_i = y_i$ for all $i <= n$.
These are finite collections of sets. You should be able to find most of the proof yourself. I would suggest proving that they're an algebra and then using something that says "finite algebras are $sigma$-algebras".
answered Aug 27 at 1:45
user24142
2,927915
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
So, $Omega$ is the infinite product $Pi_iin mathbbN H, T$. The sets you're looking at are such that if $x in A in F_n$ then $y in A$ if $x_i = y_i$ for all $i <= n$.
These are finite collections of sets. You should be able to find most of the proof yourself. I would suggest proving that they're an algebra and then using something that says "finite algebras are $sigma$-algebras".
answered Aug 27 at 1:45
user24142
2,927915
add a comment |Â
up vote
0
down vote
So, $Omega$ is the infinite product $Pi_iin mathbbN H, T$. The sets you're looking at are such that if $x in A in F_n$ then $y in A$ if $x_i = y_i$ for all $i <= n$.
These are finite collections of sets. You should be able to find most of the proof yourself. I would suggest proving that they're an algebra and then using something that says "finite algebras are $sigma$-algebras".
answered Aug 27 at 1:45
user24142
2,927915
add a comment |Â
up vote
0
down vote
up vote
0
down vote
So, $Omega$ is the infinite product $Pi_iin mathbbN H, T$. The sets you're looking at are such that if $x in A in F_n$ then $y in A$ if $x_i = y_i$ for all $i <= n$.
These are finite collections of sets. You should be able to find most of the proof yourself. I would suggest proving that they're an algebra and then using something that says "finite algebras are $sigma$-algebras".
answered Aug 27 at 1:45
user24142
2,927915
So, $Omega$ is the infinite product $Pi_iin mathbbN H, T$. The sets you're looking at are such that if $x in A in F_n$ then $y in A$ if $x_i = y_i$ for all $i <= n$.
These are finite collections of sets. You should be able to find most of the proof yourself. I would suggest proving that they're an algebra and then using something that says "finite algebras are $sigma$-algebras".
answered Aug 27 at 1:45
user24142
2,927915
answered Aug 27 at 1:45
user24142
2,927915
answered Aug 27 at 1:45
user24142
2,927915
answered Aug 27 at 1:45
user24142
2,927915
2,927915
add a comment |Â
add a comment |Â
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I don't understand what "determined by the first n tosses" means.
– DanielWainfleet
Aug 26 at 23:54
For example,Let $A_1$ be the set of all elements of Ω such that there are exactly 2 heads during the first 4 coin tosses. Then, $A_1 ∈ F_4$.
– D.S
Aug 26 at 23:56
1
This problem seems to require you to do two things. First, say exactly what $F_n$ is --- not "for example" as in your comment but a mathematically precise definition. Second, combine that and the definition of "$sigma$-algebra" to determine exactly what you need to prove. Technically, there's a third step, namely to prove those things, but I think that, once you've got the goals written down exactly, they will be very easy to prove. (P.S. I'm not saying that your "For example" comment won't clarify for @DanielWainfleet what you intended, but something more precise is needed for a proof.)
– Andreas Blass
Aug 27 at 0:29