$2N$ independent random variables. Does the sum of $N$ random variables and the sum of the other $N$ are...
$begingroup$
$2N$ independent random variables. does the sum of $N$ random variables and the some of the other $N$ are independent?$
I'm quite sure that it's true, but I don't really know how to prove that.
probability independence
asked Dec 23 '18 at 1:00
Mr.OYMr.OY
16610
$endgroup$
add a comment |
$begingroup$
$2N$ independent random variables. does the sum of $N$ random variables and the some of the other $N$ are independent?$
I'm quite sure that it's true, but I don't really know how to prove that.
probability independence
asked Dec 23 '18 at 1:00
Mr.OYMr.OY
16610
$endgroup$
add a comment |
$begingroup$
$2N$ independent random variables. does the sum of $N$ random variables and the some of the other $N$ are independent?$
I'm quite sure that it's true, but I don't really know how to prove that.
probability independence
asked Dec 23 '18 at 1:00
Mr.OYMr.OY
16610
$endgroup$
$2N$ independent random variables. does the sum of $N$ random variables and the some of the other $N$ are independent?$
I'm quite sure that it's true, but I don't really know how to prove that.
probability independence
probability independence
asked Dec 23 '18 at 1:00
Mr.OYMr.OY
16610
asked Dec 23 '18 at 1:00
Mr.OYMr.OY
16610
asked Dec 23 '18 at 1:00
Mr.OYMr.OY
16610
asked Dec 23 '18 at 1:00
Mr.OYMr.OY
16610
asked Dec 23 '18 at 1:00
Mr.OYMr.OY
16610
16610
add a comment |
add a comment |
1 Answer
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If ${X_i:1leq i leq n}cup {Y_j:1leq j leq m}$ is independent then $f(X_1,X_2,...,X_n)$ and $(Y_1,Y_2,...,Y_m)$ are independent for any Borel measaurable functions $f: mathbb R^{n} to mathbb R$ and $g: mathbb R^{m} to mathbb R$. This follows from the fact that $sigma (X_1,X_2,...,X_n)$ and $sigma (Y_1,Y_2,...,Y_m)$ are independent. [Note that $sigma (X_1,X_2,...,X_n)$ is generated by sets of the type $X_1^{-1}(A_1)cap X_2^{-1}(A_2)cap ...cap X_n^{-1}(A_n)$ where $A_i$'s are Borel sets. Similarly for $sigma (Y_1,Y_2,...,Y_m)$].
answered Dec 23 '18 at 5:18
Kavi Rama MurthyKavi Rama Murthy
61.2k42262
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
If ${X_i:1leq i leq n}cup {Y_j:1leq j leq m}$ is independent then $f(X_1,X_2,...,X_n)$ and $(Y_1,Y_2,...,Y_m)$ are independent for any Borel measaurable functions $f: mathbb R^{n} to mathbb R$ and $g: mathbb R^{m} to mathbb R$. This follows from the fact that $sigma (X_1,X_2,...,X_n)$ and $sigma (Y_1,Y_2,...,Y_m)$ are independent. [Note that $sigma (X_1,X_2,...,X_n)$ is generated by sets of the type $X_1^{-1}(A_1)cap X_2^{-1}(A_2)cap ...cap X_n^{-1}(A_n)$ where $A_i$'s are Borel sets. Similarly for $sigma (Y_1,Y_2,...,Y_m)$].
answered Dec 23 '18 at 5:18
Kavi Rama MurthyKavi Rama Murthy
61.2k42262
$endgroup$
add a comment |
$begingroup$
If ${X_i:1leq i leq n}cup {Y_j:1leq j leq m}$ is independent then $f(X_1,X_2,...,X_n)$ and $(Y_1,Y_2,...,Y_m)$ are independent for any Borel measaurable functions $f: mathbb R^{n} to mathbb R$ and $g: mathbb R^{m} to mathbb R$. This follows from the fact that $sigma (X_1,X_2,...,X_n)$ and $sigma (Y_1,Y_2,...,Y_m)$ are independent. [Note that $sigma (X_1,X_2,...,X_n)$ is generated by sets of the type $X_1^{-1}(A_1)cap X_2^{-1}(A_2)cap ...cap X_n^{-1}(A_n)$ where $A_i$'s are Borel sets. Similarly for $sigma (Y_1,Y_2,...,Y_m)$].
answered Dec 23 '18 at 5:18
Kavi Rama MurthyKavi Rama Murthy
61.2k42262
$endgroup$
add a comment |
$begingroup$
If ${X_i:1leq i leq n}cup {Y_j:1leq j leq m}$ is independent then $f(X_1,X_2,...,X_n)$ and $(Y_1,Y_2,...,Y_m)$ are independent for any Borel measaurable functions $f: mathbb R^{n} to mathbb R$ and $g: mathbb R^{m} to mathbb R$. This follows from the fact that $sigma (X_1,X_2,...,X_n)$ and $sigma (Y_1,Y_2,...,Y_m)$ are independent. [Note that $sigma (X_1,X_2,...,X_n)$ is generated by sets of the type $X_1^{-1}(A_1)cap X_2^{-1}(A_2)cap ...cap X_n^{-1}(A_n)$ where $A_i$'s are Borel sets. Similarly for $sigma (Y_1,Y_2,...,Y_m)$].
answered Dec 23 '18 at 5:18
Kavi Rama MurthyKavi Rama Murthy
61.2k42262
$endgroup$
If ${X_i:1leq i leq n}cup {Y_j:1leq j leq m}$ is independent then $f(X_1,X_2,...,X_n)$ and $(Y_1,Y_2,...,Y_m)$ are independent for any Borel measaurable functions $f: mathbb R^{n} to mathbb R$ and $g: mathbb R^{m} to mathbb R$. This follows from the fact that $sigma (X_1,X_2,...,X_n)$ and $sigma (Y_1,Y_2,...,Y_m)$ are independent. [Note that $sigma (X_1,X_2,...,X_n)$ is generated by sets of the type $X_1^{-1}(A_1)cap X_2^{-1}(A_2)cap ...cap X_n^{-1}(A_n)$ where $A_i$'s are Borel sets. Similarly for $sigma (Y_1,Y_2,...,Y_m)$].
answered Dec 23 '18 at 5:18
Kavi Rama MurthyKavi Rama Murthy
61.2k42262
edited Dec 23 '18 at 5:23
answered Dec 23 '18 at 5:18
Kavi Rama MurthyKavi Rama Murthy
61.2k42262
answered Dec 23 '18 at 5:18
Kavi Rama MurthyKavi Rama Murthy
61.2k42262
answered Dec 23 '18 at 5:18
Kavi Rama MurthyKavi Rama Murthy
61.2k42262
61.2k42262
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