Sum of two distributions
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How is the sum of two distribution defined? I find this concept applied throughout my book but I never found the precise definition...
Is it just $<Lambda_1 + Lambda_2,phi>:=<Lambda_1,phi>+<Lambda_2,phi> $?
functional-analysis distribution-theory
asked Dec 15 '18 at 9:44
LeonardoLeonardo
3339
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add a comment |
$begingroup$
How is the sum of two distribution defined? I find this concept applied throughout my book but I never found the precise definition...
Is it just $<Lambda_1 + Lambda_2,phi>:=<Lambda_1,phi>+<Lambda_2,phi> $?
functional-analysis distribution-theory
asked Dec 15 '18 at 9:44
LeonardoLeonardo
3339
$endgroup$
1
$begingroup$
I am not familiar with "sum of distributions". Could it concern the distribution of $X+Y$ where $X$ and $Y$ are two independent random variables defined on the same probability space? Here I am talking about distributions in the context of probability theory, so it might be completely irrelevant what I am saying.
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– drhab
Dec 15 '18 at 10:00
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I think he means distribution in functional analysis as the tag implies
$endgroup$
– orange
Dec 15 '18 at 10:15
add a comment |
$begingroup$
How is the sum of two distribution defined? I find this concept applied throughout my book but I never found the precise definition...
Is it just $<Lambda_1 + Lambda_2,phi>:=<Lambda_1,phi>+<Lambda_2,phi> $?
functional-analysis distribution-theory
asked Dec 15 '18 at 9:44
LeonardoLeonardo
3339
$endgroup$
How is the sum of two distribution defined? I find this concept applied throughout my book but I never found the precise definition...
Is it just $<Lambda_1 + Lambda_2,phi>:=<Lambda_1,phi>+<Lambda_2,phi> $?
functional-analysis distribution-theory
functional-analysis distribution-theory
asked Dec 15 '18 at 9:44
LeonardoLeonardo
3339
asked Dec 15 '18 at 9:44
LeonardoLeonardo
3339
asked Dec 15 '18 at 9:44
LeonardoLeonardo
3339
asked Dec 15 '18 at 9:44
LeonardoLeonardo
3339
asked Dec 15 '18 at 9:44
LeonardoLeonardo
3339
3339
1
$begingroup$
I am not familiar with "sum of distributions". Could it concern the distribution of $X+Y$ where $X$ and $Y$ are two independent random variables defined on the same probability space? Here I am talking about distributions in the context of probability theory, so it might be completely irrelevant what I am saying.
$endgroup$
– drhab
Dec 15 '18 at 10:00
$begingroup$
I think he means distribution in functional analysis as the tag implies
$endgroup$
– orange
Dec 15 '18 at 10:15
add a comment |
1
$begingroup$
I am not familiar with "sum of distributions". Could it concern the distribution of $X+Y$ where $X$ and $Y$ are two independent random variables defined on the same probability space? Here I am talking about distributions in the context of probability theory, so it might be completely irrelevant what I am saying.
$endgroup$
– drhab
Dec 15 '18 at 10:00
$begingroup$
I think he means distribution in functional analysis as the tag implies
$endgroup$
– orange
Dec 15 '18 at 10:15
1
1
$begingroup$
I am not familiar with "sum of distributions". Could it concern the distribution of $X+Y$ where $X$ and $Y$ are two independent random variables defined on the same probability space? Here I am talking about distributions in the context of probability theory, so it might be completely irrelevant what I am saying.
$endgroup$
– drhab
Dec 15 '18 at 10:00
$begingroup$
I am not familiar with "sum of distributions". Could it concern the distribution of $X+Y$ where $X$ and $Y$ are two independent random variables defined on the same probability space? Here I am talking about distributions in the context of probability theory, so it might be completely irrelevant what I am saying.
$endgroup$
– drhab
Dec 15 '18 at 10:00
$begingroup$
I think he means distribution in functional analysis as the tag implies
$endgroup$
– orange
Dec 15 '18 at 10:15
$begingroup$
I think he means distribution in functional analysis as the tag implies
$endgroup$
– orange
Dec 15 '18 at 10:15
add a comment |
1 Answer
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Yes, this is probably the most natural definition for the sum of two distributions. And this is not hard to check that the $Lambda_1+Lambda_2$ defined in this way is a distribution.
answered Dec 15 '18 at 10:03
Davide GiraudoDavide Giraudo
127k17154268
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add a comment |
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$begingroup$
Yes, this is probably the most natural definition for the sum of two distributions. And this is not hard to check that the $Lambda_1+Lambda_2$ defined in this way is a distribution.
answered Dec 15 '18 at 10:03
Davide GiraudoDavide Giraudo
127k17154268
$endgroup$
add a comment |
$begingroup$
Yes, this is probably the most natural definition for the sum of two distributions. And this is not hard to check that the $Lambda_1+Lambda_2$ defined in this way is a distribution.
answered Dec 15 '18 at 10:03
Davide GiraudoDavide Giraudo
127k17154268
$endgroup$
add a comment |
$begingroup$
Yes, this is probably the most natural definition for the sum of two distributions. And this is not hard to check that the $Lambda_1+Lambda_2$ defined in this way is a distribution.
answered Dec 15 '18 at 10:03
Davide GiraudoDavide Giraudo
127k17154268
$endgroup$
Yes, this is probably the most natural definition for the sum of two distributions. And this is not hard to check that the $Lambda_1+Lambda_2$ defined in this way is a distribution.
answered Dec 15 '18 at 10:03
Davide GiraudoDavide Giraudo
127k17154268
answered Dec 15 '18 at 10:03
Davide GiraudoDavide Giraudo
127k17154268
answered Dec 15 '18 at 10:03
Davide GiraudoDavide Giraudo
127k17154268
answered Dec 15 '18 at 10:03
Davide GiraudoDavide Giraudo
127k17154268
127k17154268
add a comment |
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1
$begingroup$
I am not familiar with "sum of distributions". Could it concern the distribution of $X+Y$ where $X$ and $Y$ are two independent random variables defined on the same probability space? Here I am talking about distributions in the context of probability theory, so it might be completely irrelevant what I am saying.
$endgroup$
– drhab
Dec 15 '18 at 10:00
$begingroup$
I think he means distribution in functional analysis as the tag implies
$endgroup$
– orange
Dec 15 '18 at 10:15