Mixing
About dependency of random variables
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I am always considering non-discrete/non-finite probability spaces $Omega$. For everything that follows feel free to assume $Omega = mathbb{R}^n$.
Say you have 2 random variables $X_1,X_2 :Omega rightarrow mathbb{R}$. Now an event based definition of $X_1$ and $X_2$ being ``independent" is as follows : "$X_1$ and $X_2$ are independent random variables if for all $x,y in mathbb{R}$ we have that $mathbb{P}((X_1 leq x)cap(X_2 leq y)) = mathbb{P}(X_1 leq x)mathbb{P}(X_2 leq y)$" (Am I right?)
Are there natural examples of pairs of independent random variables whose descriptions can be given as maps $Omega rightarrow mathbb{R}$ ?
Is the above setup enough to ensure that there exists a joint-distribution of $X_1$ and $X_2$? If yes, how?
When would one prefer to use a joint-distribution based definition of ``independence" as opposed to such an event based defition and vice-versa?
probability stochastic-processes independence
asked 5 hours ago
gradstudent
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0
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I am always considering non-discrete/non-finite probability spaces $Omega$. For everything that follows feel free to assume $Omega = mathbb{R}^n$.
Say you have 2 random variables $X_1,X_2 :Omega rightarrow mathbb{R}$. Now an event based definition of $X_1$ and $X_2$ being ``independent" is as follows : "$X_1$ and $X_2$ are independent random variables if for all $x,y in mathbb{R}$ we have that $mathbb{P}((X_1 leq x)cap(X_2 leq y)) = mathbb{P}(X_1 leq x)mathbb{P}(X_2 leq y)$" (Am I right?)
Are there natural examples of pairs of independent random variables whose descriptions can be given as maps $Omega rightarrow mathbb{R}$ ?
Is the above setup enough to ensure that there exists a joint-distribution of $X_1$ and $X_2$? If yes, how?
When would one prefer to use a joint-distribution based definition of ``independence" as opposed to such an event based defition and vice-versa?
probability stochastic-processes independence
asked 5 hours ago
gradstudent
18017
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am always considering non-discrete/non-finite probability spaces $Omega$. For everything that follows feel free to assume $Omega = mathbb{R}^n$.
Say you have 2 random variables $X_1,X_2 :Omega rightarrow mathbb{R}$. Now an event based definition of $X_1$ and $X_2$ being ``independent" is as follows : "$X_1$ and $X_2$ are independent random variables if for all $x,y in mathbb{R}$ we have that $mathbb{P}((X_1 leq x)cap(X_2 leq y)) = mathbb{P}(X_1 leq x)mathbb{P}(X_2 leq y)$" (Am I right?)
Are there natural examples of pairs of independent random variables whose descriptions can be given as maps $Omega rightarrow mathbb{R}$ ?
Is the above setup enough to ensure that there exists a joint-distribution of $X_1$ and $X_2$? If yes, how?
When would one prefer to use a joint-distribution based definition of ``independence" as opposed to such an event based defition and vice-versa?
probability stochastic-processes independence
asked 5 hours ago
gradstudent
18017
I am always considering non-discrete/non-finite probability spaces $Omega$. For everything that follows feel free to assume $Omega = mathbb{R}^n$.
Say you have 2 random variables $X_1,X_2 :Omega rightarrow mathbb{R}$. Now an event based definition of $X_1$ and $X_2$ being ``independent" is as follows : "$X_1$ and $X_2$ are independent random variables if for all $x,y in mathbb{R}$ we have that $mathbb{P}((X_1 leq x)cap(X_2 leq y)) = mathbb{P}(X_1 leq x)mathbb{P}(X_2 leq y)$" (Am I right?)
Are there natural examples of pairs of independent random variables whose descriptions can be given as maps $Omega rightarrow mathbb{R}$ ?
Is the above setup enough to ensure that there exists a joint-distribution of $X_1$ and $X_2$? If yes, how?
When would one prefer to use a joint-distribution based definition of ``independence" as opposed to such an event based defition and vice-versa?
probability stochastic-processes independence
probability stochastic-processes independence
asked 5 hours ago
gradstudent
18017
asked 5 hours ago
gradstudent
18017
edited 4 hours ago
asked 5 hours ago
gradstudent
18017
asked 5 hours ago
gradstudent
18017
asked 5 hours ago
gradstudent
18017
18017
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1 Answer
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On $(0,1)$ if you define $X_n(omega)$ as the n-th coefficient in the expansion of $omega$ to base $2$ then the random variables $X_1,X_2,cdots$ are independent. The statement $P{Xleq x,Yleq y}=P{Xleq x}P{Yleq y}$ for all $x,y$ is equivalent to the statement $P(Xin A, Y in B)=P(Xin A)P( Y in B)$ for all Borel sets $A$ and $B$ in $mathbb R$. (I suppose this is what you mean by event based definition of independence).
answered 4 hours ago
Kavi Rama Murthy
39.9k31750
Thanks for your reply! (1) I was looking for "natural" example of independent random variables which can be described as such $Omega rightarrow mathbb{R}$. Could you kindly provides some? Say some examples in terms of random variables with commonly used distributions like Gaussian, chi^2 etc.. (2) I agree that the definition I gave in the question is the event based definition. I wanted to know when can one give a joint distribution based definition and when is it more or less useful. (In my setup is a joint-distribution of $X_1$ and $X_2$ guaranteed to exist?)
– gradstudent
4 hours ago
On $(0,1)times (0,1)$ the coordinate functions $(x,y) to x$ and $(x,y) to y$ are independent random variables.
– Kavi Rama Murthy
4 hours ago
Well, the $X_1$ and $X_2$ that you gave are not random variables with very "standard" distributions. Thats what makes them "unnatural" in my view. I am looking for situations which can be specified as maps and the distributions of the two random variables is something "nice" like Gaussian etc.
– gradstudent
4 hours ago
Both have uniform distribution on $(0,1)$.
– Kavi Rama Murthy
4 hours ago
hmm. Can one lets say give examples of $X_1$ and $X_2$ maps n this form such that the joint distribution of $(X_1,X_2)$ is the 2-dimensional standard Gaussian?
– gradstudent
4 hours ago
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
On $(0,1)$ if you define $X_n(omega)$ as the n-th coefficient in the expansion of $omega$ to base $2$ then the random variables $X_1,X_2,cdots$ are independent. The statement $P{Xleq x,Yleq y}=P{Xleq x}P{Yleq y}$ for all $x,y$ is equivalent to the statement $P(Xin A, Y in B)=P(Xin A)P( Y in B)$ for all Borel sets $A$ and $B$ in $mathbb R$. (I suppose this is what you mean by event based definition of independence).
answered 4 hours ago
Kavi Rama Murthy
39.9k31750
Thanks for your reply! (1) I was looking for "natural" example of independent random variables which can be described as such $Omega rightarrow mathbb{R}$. Could you kindly provides some? Say some examples in terms of random variables with commonly used distributions like Gaussian, chi^2 etc.. (2) I agree that the definition I gave in the question is the event based definition. I wanted to know when can one give a joint distribution based definition and when is it more or less useful. (In my setup is a joint-distribution of $X_1$ and $X_2$ guaranteed to exist?)
– gradstudent
4 hours ago
On $(0,1)times (0,1)$ the coordinate functions $(x,y) to x$ and $(x,y) to y$ are independent random variables.
– Kavi Rama Murthy
4 hours ago
Well, the $X_1$ and $X_2$ that you gave are not random variables with very "standard" distributions. Thats what makes them "unnatural" in my view. I am looking for situations which can be specified as maps and the distributions of the two random variables is something "nice" like Gaussian etc.
– gradstudent
4 hours ago
Both have uniform distribution on $(0,1)$.
– Kavi Rama Murthy
4 hours ago
hmm. Can one lets say give examples of $X_1$ and $X_2$ maps n this form such that the joint distribution of $(X_1,X_2)$ is the 2-dimensional standard Gaussian?
– gradstudent
4 hours ago
add a comment |
up vote
0
down vote
On $(0,1)$ if you define $X_n(omega)$ as the n-th coefficient in the expansion of $omega$ to base $2$ then the random variables $X_1,X_2,cdots$ are independent. The statement $P{Xleq x,Yleq y}=P{Xleq x}P{Yleq y}$ for all $x,y$ is equivalent to the statement $P(Xin A, Y in B)=P(Xin A)P( Y in B)$ for all Borel sets $A$ and $B$ in $mathbb R$. (I suppose this is what you mean by event based definition of independence).
answered 4 hours ago
Kavi Rama Murthy
39.9k31750
Thanks for your reply! (1) I was looking for "natural" example of independent random variables which can be described as such $Omega rightarrow mathbb{R}$. Could you kindly provides some? Say some examples in terms of random variables with commonly used distributions like Gaussian, chi^2 etc.. (2) I agree that the definition I gave in the question is the event based definition. I wanted to know when can one give a joint distribution based definition and when is it more or less useful. (In my setup is a joint-distribution of $X_1$ and $X_2$ guaranteed to exist?)
– gradstudent
4 hours ago
On $(0,1)times (0,1)$ the coordinate functions $(x,y) to x$ and $(x,y) to y$ are independent random variables.
– Kavi Rama Murthy
4 hours ago
Well, the $X_1$ and $X_2$ that you gave are not random variables with very "standard" distributions. Thats what makes them "unnatural" in my view. I am looking for situations which can be specified as maps and the distributions of the two random variables is something "nice" like Gaussian etc.
– gradstudent
4 hours ago
Both have uniform distribution on $(0,1)$.
– Kavi Rama Murthy
4 hours ago
hmm. Can one lets say give examples of $X_1$ and $X_2$ maps n this form such that the joint distribution of $(X_1,X_2)$ is the 2-dimensional standard Gaussian?
– gradstudent
4 hours ago
add a comment |
up vote
0
down vote
up vote
0
down vote
On $(0,1)$ if you define $X_n(omega)$ as the n-th coefficient in the expansion of $omega$ to base $2$ then the random variables $X_1,X_2,cdots$ are independent. The statement $P{Xleq x,Yleq y}=P{Xleq x}P{Yleq y}$ for all $x,y$ is equivalent to the statement $P(Xin A, Y in B)=P(Xin A)P( Y in B)$ for all Borel sets $A$ and $B$ in $mathbb R$. (I suppose this is what you mean by event based definition of independence).
answered 4 hours ago
Kavi Rama Murthy
39.9k31750
On $(0,1)$ if you define $X_n(omega)$ as the n-th coefficient in the expansion of $omega$ to base $2$ then the random variables $X_1,X_2,cdots$ are independent. The statement $P{Xleq x,Yleq y}=P{Xleq x}P{Yleq y}$ for all $x,y$ is equivalent to the statement $P(Xin A, Y in B)=P(Xin A)P( Y in B)$ for all Borel sets $A$ and $B$ in $mathbb R$. (I suppose this is what you mean by event based definition of independence).
answered 4 hours ago
Kavi Rama Murthy
39.9k31750
answered 4 hours ago
Kavi Rama Murthy
39.9k31750
answered 4 hours ago
Kavi Rama Murthy
39.9k31750
answered 4 hours ago
Kavi Rama Murthy
39.9k31750
39.9k31750
Thanks for your reply! (1) I was looking for "natural" example of independent random variables which can be described as such $Omega rightarrow mathbb{R}$. Could you kindly provides some? Say some examples in terms of random variables with commonly used distributions like Gaussian, chi^2 etc.. (2) I agree that the definition I gave in the question is the event based definition. I wanted to know when can one give a joint distribution based definition and when is it more or less useful. (In my setup is a joint-distribution of $X_1$ and $X_2$ guaranteed to exist?)
– gradstudent
4 hours ago
On $(0,1)times (0,1)$ the coordinate functions $(x,y) to x$ and $(x,y) to y$ are independent random variables.
– Kavi Rama Murthy
4 hours ago
Well, the $X_1$ and $X_2$ that you gave are not random variables with very "standard" distributions. Thats what makes them "unnatural" in my view. I am looking for situations which can be specified as maps and the distributions of the two random variables is something "nice" like Gaussian etc.
– gradstudent
4 hours ago
Both have uniform distribution on $(0,1)$.
– Kavi Rama Murthy
4 hours ago
hmm. Can one lets say give examples of $X_1$ and $X_2$ maps n this form such that the joint distribution of $(X_1,X_2)$ is the 2-dimensional standard Gaussian?
– gradstudent
4 hours ago
add a comment |
Thanks for your reply! (1) I was looking for "natural" example of independent random variables which can be described as such $Omega rightarrow mathbb{R}$. Could you kindly provides some? Say some examples in terms of random variables with commonly used distributions like Gaussian, chi^2 etc.. (2) I agree that the definition I gave in the question is the event based definition. I wanted to know when can one give a joint distribution based definition and when is it more or less useful. (In my setup is a joint-distribution of $X_1$ and $X_2$ guaranteed to exist?)
– gradstudent
4 hours ago
On $(0,1)times (0,1)$ the coordinate functions $(x,y) to x$ and $(x,y) to y$ are independent random variables.
– Kavi Rama Murthy
4 hours ago
Well, the $X_1$ and $X_2$ that you gave are not random variables with very "standard" distributions. Thats what makes them "unnatural" in my view. I am looking for situations which can be specified as maps and the distributions of the two random variables is something "nice" like Gaussian etc.
– gradstudent
4 hours ago
Both have uniform distribution on $(0,1)$.
– Kavi Rama Murthy
4 hours ago
hmm. Can one lets say give examples of $X_1$ and $X_2$ maps n this form such that the joint distribution of $(X_1,X_2)$ is the 2-dimensional standard Gaussian?
– gradstudent
4 hours ago
Thanks for your reply! (1) I was looking for "natural" example of independent random variables which can be described as such $Omega rightarrow mathbb{R}$. Could you kindly provides some? Say some examples in terms of random variables with commonly used distributions like Gaussian, chi^2 etc.. (2) I agree that the definition I gave in the question is the event based definition. I wanted to know when can one give a joint distribution based definition and when is it more or less useful. (In my setup is a joint-distribution of $X_1$ and $X_2$ guaranteed to exist?)
– gradstudent
4 hours ago
Thanks for your reply! (1) I was looking for "natural" example of independent random variables which can be described as such $Omega rightarrow mathbb{R}$. Could you kindly provides some? Say some examples in terms of random variables with commonly used distributions like Gaussian, chi^2 etc.. (2) I agree that the definition I gave in the question is the event based definition. I wanted to know when can one give a joint distribution based definition and when is it more or less useful. (In my setup is a joint-distribution of $X_1$ and $X_2$ guaranteed to exist?)
– gradstudent
4 hours ago
On $(0,1)times (0,1)$ the coordinate functions $(x,y) to x$ and $(x,y) to y$ are independent random variables.
– Kavi Rama Murthy
4 hours ago
On $(0,1)times (0,1)$ the coordinate functions $(x,y) to x$ and $(x,y) to y$ are independent random variables.
– Kavi Rama Murthy
4 hours ago
Well, the $X_1$ and $X_2$ that you gave are not random variables with very "standard" distributions. Thats what makes them "unnatural" in my view. I am looking for situations which can be specified as maps and the distributions of the two random variables is something "nice" like Gaussian etc.
– gradstudent
4 hours ago
Well, the $X_1$ and $X_2$ that you gave are not random variables with very "standard" distributions. Thats what makes them "unnatural" in my view. I am looking for situations which can be specified as maps and the distributions of the two random variables is something "nice" like Gaussian etc.
– gradstudent
4 hours ago
Both have uniform distribution on $(0,1)$.
– Kavi Rama Murthy
4 hours ago
Both have uniform distribution on $(0,1)$.
– Kavi Rama Murthy
4 hours ago
hmm. Can one lets say give examples of $X_1$ and $X_2$ maps n this form such that the joint distribution of $(X_1,X_2)$ is the 2-dimensional standard Gaussian?
– gradstudent
4 hours ago
hmm. Can one lets say give examples of $X_1$ and $X_2$ maps n this form such that the joint distribution of $(X_1,X_2)$ is the 2-dimensional standard Gaussian?
– gradstudent
4 hours ago
add a comment |
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