Are $sigma$ algebras of multiple coin tosses subsets?
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If we consider the $sigma$-algebra of a single coin toss, it is denoted by $Delta_1=,H,T,H,T$.
Similarly for a two coin toss it would be $Delta_2=,HH,TT,........,HH,HT,TT,TH$.
Then can it be said that $Delta_1 subset Delta_2$?
probability-theory measure-theory
edited Aug 10 at 16:34
aidangallagher4
6411312
asked Aug 10 at 16:14
Sumit
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If we consider the $sigma$-algebra of a single coin toss, it is denoted by $Delta_1=,H,T,H,T$.
Similarly for a two coin toss it would be $Delta_2=,HH,TT,........,HH,HT,TT,TH$.
Then can it be said that $Delta_1 subset Delta_2$?
probability-theory measure-theory
edited Aug 10 at 16:34
aidangallagher4
6411312
asked Aug 10 at 16:14
Sumit
1033
As a general rule of thumb, try to solve the problem before writing about it here.
– Cleric
Aug 10 at 16:42
A property which is somewhat similar to what you're talking about is that if $X$ and $Y$ are two random variables, then $sigma(X)subseteqsigma(X, Y)$. In your case $sigma(X)$ would be $HH, HT, TH, TT, emptyset, Omega$, which is in some sense isomorphic to $H, T, emptyset, Omega$.
– Jack M
Aug 10 at 16:53
But isn't this supposed to be a general property as I see from this link: nptel.ac.in/courses/108106083/…
– Sumit
Aug 10 at 17:03
@Sumit it depends on the question how you construct the $sigma$-algebra of events. It can be done is such a way that $Delta_1subsetDelta_2$, but in your question you did not do that.
– drhab
Aug 10 at 17:08
@drhab could you show me an example?
– Sumit
Aug 10 at 17:09
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
If we consider the $sigma$-algebra of a single coin toss, it is denoted by $Delta_1=,H,T,H,T$.
Similarly for a two coin toss it would be $Delta_2=,HH,TT,........,HH,HT,TT,TH$.
Then can it be said that $Delta_1 subset Delta_2$?
probability-theory measure-theory
edited Aug 10 at 16:34
aidangallagher4
6411312
asked Aug 10 at 16:14
Sumit
1033
If we consider the $sigma$-algebra of a single coin toss, it is denoted by $Delta_1=,H,T,H,T$.
Similarly for a two coin toss it would be $Delta_2=,HH,TT,........,HH,HT,TT,TH$.
Then can it be said that $Delta_1 subset Delta_2$?
probability-theory measure-theory
edited Aug 10 at 16:34
aidangallagher4
6411312
asked Aug 10 at 16:14
Sumit
1033
edited Aug 10 at 16:34
aidangallagher4
6411312
edited Aug 10 at 16:34
aidangallagher4
6411312
edited Aug 10 at 16:34
aidangallagher4
6411312
6411312
asked Aug 10 at 16:14
Sumit
1033
asked Aug 10 at 16:14
Sumit
1033
asked Aug 10 at 16:14
Sumit
1033
1033
As a general rule of thumb, try to solve the problem before writing about it here.
– Cleric
Aug 10 at 16:42
A property which is somewhat similar to what you're talking about is that if $X$ and $Y$ are two random variables, then $sigma(X)subseteqsigma(X, Y)$. In your case $sigma(X)$ would be $HH, HT, TH, TT, emptyset, Omega$, which is in some sense isomorphic to $H, T, emptyset, Omega$.
– Jack M
Aug 10 at 16:53
But isn't this supposed to be a general property as I see from this link: nptel.ac.in/courses/108106083/…
– Sumit
Aug 10 at 17:03
@Sumit it depends on the question how you construct the $sigma$-algebra of events. It can be done is such a way that $Delta_1subsetDelta_2$, but in your question you did not do that.
– drhab
Aug 10 at 17:08
@drhab could you show me an example?
– Sumit
Aug 10 at 17:09
add a comment |Â
As a general rule of thumb, try to solve the problem before writing about it here.
– Cleric
Aug 10 at 16:42
A property which is somewhat similar to what you're talking about is that if $X$ and $Y$ are two random variables, then $sigma(X)subseteqsigma(X, Y)$. In your case $sigma(X)$ would be $HH, HT, TH, TT, emptyset, Omega$, which is in some sense isomorphic to $H, T, emptyset, Omega$.
– Jack M
Aug 10 at 16:53
But isn't this supposed to be a general property as I see from this link: nptel.ac.in/courses/108106083/…
– Sumit
Aug 10 at 17:03
@Sumit it depends on the question how you construct the $sigma$-algebra of events. It can be done is such a way that $Delta_1subsetDelta_2$, but in your question you did not do that.
– drhab
Aug 10 at 17:08
@drhab could you show me an example?
– Sumit
Aug 10 at 17:09
As a general rule of thumb, try to solve the problem before writing about it here.
– Cleric
Aug 10 at 16:42
As a general rule of thumb, try to solve the problem before writing about it here.
– Cleric
Aug 10 at 16:42
A property which is somewhat similar to what you're talking about is that if $X$ and $Y$ are two random variables, then $sigma(X)subseteqsigma(X, Y)$. In your case $sigma(X)$ would be $HH, HT, TH, TT, emptyset, Omega$, which is in some sense isomorphic to $H, T, emptyset, Omega$.
– Jack M
Aug 10 at 16:53
A property which is somewhat similar to what you're talking about is that if $X$ and $Y$ are two random variables, then $sigma(X)subseteqsigma(X, Y)$. In your case $sigma(X)$ would be $HH, HT, TH, TT, emptyset, Omega$, which is in some sense isomorphic to $H, T, emptyset, Omega$.
– Jack M
Aug 10 at 16:53
But isn't this supposed to be a general property as I see from this link: nptel.ac.in/courses/108106083/…
– Sumit
Aug 10 at 17:03
But isn't this supposed to be a general property as I see from this link: nptel.ac.in/courses/108106083/…
– Sumit
Aug 10 at 17:03
@Sumit it depends on the question how you construct the $sigma$-algebra of events. It can be done is such a way that $Delta_1subsetDelta_2$, but in your question you did not do that.
– drhab
Aug 10 at 17:08
@Sumit it depends on the question how you construct the $sigma$-algebra of events. It can be done is such a way that $Delta_1subsetDelta_2$, but in your question you did not do that.
– drhab
Aug 10 at 17:08
@drhab could you show me an example?
– Sumit
Aug 10 at 17:09
@drhab could you show me an example?
– Sumit
Aug 10 at 17:09
add a comment |Â
3 Answers
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1
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accepted
Clearly not. $HinDelta_1,HnotinDelta_2 $
EDIT
In answer to the OP's question, $mathscr F_n$ is the collection of subsets of all sequences of tosses, where membership can be decided on the basis of the first $n$ tosses. For example, members of $mathscr F_1$ include the set of sequences that start with heads, the set of sequences that start with tails and so on. Clearly, if membership can be decided on the basis of the first toss, then it can be decided on the basis of the first two tosses. The set of sequences that start with two heads is a subset of the set of sequences that start with one head.
Not every sequence belongs to $mathscr F_n$ for some $n.$ Consider for example, the set of sequences with only finitely many heads., or the set of sequences where the cumulative number of tails is always greater than the cumulative number of heads.
answered Aug 10 at 16:18
saulspatz
10.8k21323
Agreed, but given a $sigma$ algebra on $omega=H,T^inf$, if $Delta$ is the collection of subsets of $omega$, then there is a property that $Delta_nsubset Delta_n+1$. What am I interpreting incorrectly?
– Sumit
Aug 10 at 16:42
@Sumit, where did you find that property?
– Cleric
Aug 10 at 16:46
In nptel.ac.in/courses/108106083/…
– Sumit
Aug 10 at 16:47
You misunderstand $mathcal F_n$. Start with an easy case; look at $mathcal F_2$, following the definition in that link.
– Cleric
Aug 10 at 17:27
add a comment |Â
up vote
3
down vote
Of course not, outcome $H$ is not in $Delta _2$.
add a comment |Â
up vote
1
down vote
Example on request (see comments on question).
Going for $2$ tosses we can take as sample space the set $Omega=HH,HT,TH,TT$ and as $sigma$-algebra the collection $Delta_2=wp(Omega)$.
In this model we focus on the first toss.
It induces the following sub-$sigma$-algebra: $$Delta_1=varnothing,HH,HT,TH,TT,OmegasubsetDelta_2$$
Note that e.g. $HH,HT$ is exactly the event that the first toss shows a head.
answered Aug 10 at 17:17
drhab
87.1k541118
add a comment |Â
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Clearly not. $HinDelta_1,HnotinDelta_2 $
EDIT
In answer to the OP's question, $mathscr F_n$ is the collection of subsets of all sequences of tosses, where membership can be decided on the basis of the first $n$ tosses. For example, members of $mathscr F_1$ include the set of sequences that start with heads, the set of sequences that start with tails and so on. Clearly, if membership can be decided on the basis of the first toss, then it can be decided on the basis of the first two tosses. The set of sequences that start with two heads is a subset of the set of sequences that start with one head.
Not every sequence belongs to $mathscr F_n$ for some $n.$ Consider for example, the set of sequences with only finitely many heads., or the set of sequences where the cumulative number of tails is always greater than the cumulative number of heads.
answered Aug 10 at 16:18
saulspatz
10.8k21323
Agreed, but given a $sigma$ algebra on $omega=H,T^inf$, if $Delta$ is the collection of subsets of $omega$, then there is a property that $Delta_nsubset Delta_n+1$. What am I interpreting incorrectly?
– Sumit
Aug 10 at 16:42
@Sumit, where did you find that property?
– Cleric
Aug 10 at 16:46
In nptel.ac.in/courses/108106083/…
– Sumit
Aug 10 at 16:47
You misunderstand $mathcal F_n$. Start with an easy case; look at $mathcal F_2$, following the definition in that link.
– Cleric
Aug 10 at 17:27
add a comment |Â
up vote
1
down vote
accepted
Clearly not. $HinDelta_1,HnotinDelta_2 $
EDIT
In answer to the OP's question, $mathscr F_n$ is the collection of subsets of all sequences of tosses, where membership can be decided on the basis of the first $n$ tosses. For example, members of $mathscr F_1$ include the set of sequences that start with heads, the set of sequences that start with tails and so on. Clearly, if membership can be decided on the basis of the first toss, then it can be decided on the basis of the first two tosses. The set of sequences that start with two heads is a subset of the set of sequences that start with one head.
Not every sequence belongs to $mathscr F_n$ for some $n.$ Consider for example, the set of sequences with only finitely many heads., or the set of sequences where the cumulative number of tails is always greater than the cumulative number of heads.
answered Aug 10 at 16:18
saulspatz
10.8k21323
Agreed, but given a $sigma$ algebra on $omega=H,T^inf$, if $Delta$ is the collection of subsets of $omega$, then there is a property that $Delta_nsubset Delta_n+1$. What am I interpreting incorrectly?
– Sumit
Aug 10 at 16:42
@Sumit, where did you find that property?
– Cleric
Aug 10 at 16:46
In nptel.ac.in/courses/108106083/…
– Sumit
Aug 10 at 16:47
You misunderstand $mathcal F_n$. Start with an easy case; look at $mathcal F_2$, following the definition in that link.
– Cleric
Aug 10 at 17:27
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Clearly not. $HinDelta_1,HnotinDelta_2 $
EDIT
In answer to the OP's question, $mathscr F_n$ is the collection of subsets of all sequences of tosses, where membership can be decided on the basis of the first $n$ tosses. For example, members of $mathscr F_1$ include the set of sequences that start with heads, the set of sequences that start with tails and so on. Clearly, if membership can be decided on the basis of the first toss, then it can be decided on the basis of the first two tosses. The set of sequences that start with two heads is a subset of the set of sequences that start with one head.
Not every sequence belongs to $mathscr F_n$ for some $n.$ Consider for example, the set of sequences with only finitely many heads., or the set of sequences where the cumulative number of tails is always greater than the cumulative number of heads.
answered Aug 10 at 16:18
saulspatz
10.8k21323
Clearly not. $HinDelta_1,HnotinDelta_2 $
EDIT
In answer to the OP's question, $mathscr F_n$ is the collection of subsets of all sequences of tosses, where membership can be decided on the basis of the first $n$ tosses. For example, members of $mathscr F_1$ include the set of sequences that start with heads, the set of sequences that start with tails and so on. Clearly, if membership can be decided on the basis of the first toss, then it can be decided on the basis of the first two tosses. The set of sequences that start with two heads is a subset of the set of sequences that start with one head.
Not every sequence belongs to $mathscr F_n$ for some $n.$ Consider for example, the set of sequences with only finitely many heads., or the set of sequences where the cumulative number of tails is always greater than the cumulative number of heads.
answered Aug 10 at 16:18
saulspatz
10.8k21323
edited Aug 10 at 17:39
answered Aug 10 at 16:18
saulspatz
10.8k21323
answered Aug 10 at 16:18
saulspatz
10.8k21323
answered Aug 10 at 16:18
saulspatz
10.8k21323
10.8k21323
Agreed, but given a $sigma$ algebra on $omega=H,T^inf$, if $Delta$ is the collection of subsets of $omega$, then there is a property that $Delta_nsubset Delta_n+1$. What am I interpreting incorrectly?
– Sumit
Aug 10 at 16:42
@Sumit, where did you find that property?
– Cleric
Aug 10 at 16:46
In nptel.ac.in/courses/108106083/…
– Sumit
Aug 10 at 16:47
You misunderstand $mathcal F_n$. Start with an easy case; look at $mathcal F_2$, following the definition in that link.
– Cleric
Aug 10 at 17:27
add a comment |Â
Agreed, but given a $sigma$ algebra on $omega=H,T^inf$, if $Delta$ is the collection of subsets of $omega$, then there is a property that $Delta_nsubset Delta_n+1$. What am I interpreting incorrectly?
– Sumit
Aug 10 at 16:42
@Sumit, where did you find that property?
– Cleric
Aug 10 at 16:46
In nptel.ac.in/courses/108106083/…
– Sumit
Aug 10 at 16:47
You misunderstand $mathcal F_n$. Start with an easy case; look at $mathcal F_2$, following the definition in that link.
– Cleric
Aug 10 at 17:27
Agreed, but given a $sigma$ algebra on $omega=H,T^inf$, if $Delta$ is the collection of subsets of $omega$, then there is a property that $Delta_nsubset Delta_n+1$. What am I interpreting incorrectly?
– Sumit
Aug 10 at 16:42
Agreed, but given a $sigma$ algebra on $omega=H,T^inf$, if $Delta$ is the collection of subsets of $omega$, then there is a property that $Delta_nsubset Delta_n+1$. What am I interpreting incorrectly?
– Sumit
Aug 10 at 16:42
@Sumit, where did you find that property?
– Cleric
Aug 10 at 16:46
@Sumit, where did you find that property?
– Cleric
Aug 10 at 16:46
In nptel.ac.in/courses/108106083/…
– Sumit
Aug 10 at 16:47
In nptel.ac.in/courses/108106083/…
– Sumit
Aug 10 at 16:47
You misunderstand $mathcal F_n$. Start with an easy case; look at $mathcal F_2$, following the definition in that link.
– Cleric
Aug 10 at 17:27
You misunderstand $mathcal F_n$. Start with an easy case; look at $mathcal F_2$, following the definition in that link.
– Cleric
Aug 10 at 17:27
add a comment |Â
up vote
3
down vote
Of course not, outcome $H$ is not in $Delta _2$.
add a comment |Â
up vote
3
down vote
Of course not, outcome $H$ is not in $Delta _2$.
add a comment |Â
up vote
3
down vote
up vote
3
down vote
Of course not, outcome $H$ is not in $Delta _2$.
Of course not, outcome $H$ is not in $Delta _2$.
answered Aug 10 at 16:17
user582949
add a comment |Â
add a comment |Â
up vote
1
down vote
Example on request (see comments on question).
Going for $2$ tosses we can take as sample space the set $Omega=HH,HT,TH,TT$ and as $sigma$-algebra the collection $Delta_2=wp(Omega)$.
In this model we focus on the first toss.
It induces the following sub-$sigma$-algebra: $$Delta_1=varnothing,HH,HT,TH,TT,OmegasubsetDelta_2$$
Note that e.g. $HH,HT$ is exactly the event that the first toss shows a head.
answered Aug 10 at 17:17
drhab
87.1k541118
add a comment |Â
up vote
1
down vote
Example on request (see comments on question).
Going for $2$ tosses we can take as sample space the set $Omega=HH,HT,TH,TT$ and as $sigma$-algebra the collection $Delta_2=wp(Omega)$.
In this model we focus on the first toss.
It induces the following sub-$sigma$-algebra: $$Delta_1=varnothing,HH,HT,TH,TT,OmegasubsetDelta_2$$
Note that e.g. $HH,HT$ is exactly the event that the first toss shows a head.
answered Aug 10 at 17:17
drhab
87.1k541118
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Example on request (see comments on question).
Going for $2$ tosses we can take as sample space the set $Omega=HH,HT,TH,TT$ and as $sigma$-algebra the collection $Delta_2=wp(Omega)$.
In this model we focus on the first toss.
It induces the following sub-$sigma$-algebra: $$Delta_1=varnothing,HH,HT,TH,TT,OmegasubsetDelta_2$$
Note that e.g. $HH,HT$ is exactly the event that the first toss shows a head.
answered Aug 10 at 17:17
drhab
87.1k541118
Example on request (see comments on question).
Going for $2$ tosses we can take as sample space the set $Omega=HH,HT,TH,TT$ and as $sigma$-algebra the collection $Delta_2=wp(Omega)$.
In this model we focus on the first toss.
It induces the following sub-$sigma$-algebra: $$Delta_1=varnothing,HH,HT,TH,TT,OmegasubsetDelta_2$$
Note that e.g. $HH,HT$ is exactly the event that the first toss shows a head.
answered Aug 10 at 17:17
drhab
87.1k541118
edited Aug 10 at 17:31
answered Aug 10 at 17:17
drhab
87.1k541118
answered Aug 10 at 17:17
drhab
87.1k541118
answered Aug 10 at 17:17
drhab
87.1k541118
87.1k541118
add a comment |Â
add a comment |Â
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As a general rule of thumb, try to solve the problem before writing about it here.
– Cleric
Aug 10 at 16:42
A property which is somewhat similar to what you're talking about is that if $X$ and $Y$ are two random variables, then $sigma(X)subseteqsigma(X, Y)$. In your case $sigma(X)$ would be $HH, HT, TH, TT, emptyset, Omega$, which is in some sense isomorphic to $H, T, emptyset, Omega$.
– Jack M
Aug 10 at 16:53
But isn't this supposed to be a general property as I see from this link: nptel.ac.in/courses/108106083/…
– Sumit
Aug 10 at 17:03
@Sumit it depends on the question how you construct the $sigma$-algebra of events. It can be done is such a way that $Delta_1subsetDelta_2$, but in your question you did not do that.
– drhab
Aug 10 at 17:08
@drhab could you show me an example?
– Sumit
Aug 10 at 17:09