Are all homogeneous functions of degree 1 of the form F(x)=k.x where k is any real number?
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I was having a look at the derivation of the PDF of the Normal Distribution by Gauss (or supposedly Gauss' derivation) and at some point a homogeneuos function of degree one shows up and Gauss infers that a possible function that satisfies that functional equation is F(x) = k.x so then he can set this famous differential equation whose solution is the PDF of the Normal Distribution: (dF/dx)/(x-mean).F = -k , where k is some real number. I am curious to know if this is the only solution or a possible solution among many others, because if that were the case then there could be another PDF for the Normal. I'll leave you the explanation here so you can judge it by yourself. https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/stahl96.pdf Page 10 onwards.
probability functional-analysis statistics
asked Nov 24 '18 at 21:43
Juan123Juan123
815
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add a comment |
$begingroup$
I was having a look at the derivation of the PDF of the Normal Distribution by Gauss (or supposedly Gauss' derivation) and at some point a homogeneuos function of degree one shows up and Gauss infers that a possible function that satisfies that functional equation is F(x) = k.x so then he can set this famous differential equation whose solution is the PDF of the Normal Distribution: (dF/dx)/(x-mean).F = -k , where k is some real number. I am curious to know if this is the only solution or a possible solution among many others, because if that were the case then there could be another PDF for the Normal. I'll leave you the explanation here so you can judge it by yourself. https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/stahl96.pdf Page 10 onwards.
probability functional-analysis statistics
asked Nov 24 '18 at 21:43
Juan123Juan123
815
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What does a "homogeneous function of degree one" means?
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– Will M.
Nov 24 '18 at 22:43
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A homogeneous function of degree 1 means that a function satisfies this property F(x.k)=k.F(x), k is a constant. A homogeneous function of degree n is F(x.k)=k^n . F(x)
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– Juan123
Nov 24 '18 at 22:58
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Did you noticed then that $F(x)=xF(1)$ and so, $F$ is the function $y = kx$ with $k = F(1)$? Either you completely lost it, or else, I am not getting it.
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– Will M.
Nov 25 '18 at 2:34
add a comment |
$begingroup$
I was having a look at the derivation of the PDF of the Normal Distribution by Gauss (or supposedly Gauss' derivation) and at some point a homogeneuos function of degree one shows up and Gauss infers that a possible function that satisfies that functional equation is F(x) = k.x so then he can set this famous differential equation whose solution is the PDF of the Normal Distribution: (dF/dx)/(x-mean).F = -k , where k is some real number. I am curious to know if this is the only solution or a possible solution among many others, because if that were the case then there could be another PDF for the Normal. I'll leave you the explanation here so you can judge it by yourself. https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/stahl96.pdf Page 10 onwards.
probability functional-analysis statistics
asked Nov 24 '18 at 21:43
Juan123Juan123
815
$endgroup$
I was having a look at the derivation of the PDF of the Normal Distribution by Gauss (or supposedly Gauss' derivation) and at some point a homogeneuos function of degree one shows up and Gauss infers that a possible function that satisfies that functional equation is F(x) = k.x so then he can set this famous differential equation whose solution is the PDF of the Normal Distribution: (dF/dx)/(x-mean).F = -k , where k is some real number. I am curious to know if this is the only solution or a possible solution among many others, because if that were the case then there could be another PDF for the Normal. I'll leave you the explanation here so you can judge it by yourself. https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/stahl96.pdf Page 10 onwards.
probability functional-analysis statistics
probability functional-analysis statistics
asked Nov 24 '18 at 21:43
Juan123Juan123
815
asked Nov 24 '18 at 21:43
Juan123Juan123
815
asked Nov 24 '18 at 21:43
Juan123Juan123
815
asked Nov 24 '18 at 21:43
Juan123Juan123
815
asked Nov 24 '18 at 21:43
Juan123Juan123
815
815
$begingroup$
What does a "homogeneous function of degree one" means?
$endgroup$
– Will M.
Nov 24 '18 at 22:43
$begingroup$
A homogeneous function of degree 1 means that a function satisfies this property F(x.k)=k.F(x), k is a constant. A homogeneous function of degree n is F(x.k)=k^n . F(x)
$endgroup$
– Juan123
Nov 24 '18 at 22:58
$begingroup$
Did you noticed then that $F(x)=xF(1)$ and so, $F$ is the function $y = kx$ with $k = F(1)$? Either you completely lost it, or else, I am not getting it.
$endgroup$
– Will M.
Nov 25 '18 at 2:34
add a comment |
$begingroup$
What does a "homogeneous function of degree one" means?
$endgroup$
– Will M.
Nov 24 '18 at 22:43
$begingroup$
A homogeneous function of degree 1 means that a function satisfies this property F(x.k)=k.F(x), k is a constant. A homogeneous function of degree n is F(x.k)=k^n . F(x)
$endgroup$
– Juan123
Nov 24 '18 at 22:58
$begingroup$
Did you noticed then that $F(x)=xF(1)$ and so, $F$ is the function $y = kx$ with $k = F(1)$? Either you completely lost it, or else, I am not getting it.
$endgroup$
– Will M.
Nov 25 '18 at 2:34
$begingroup$
What does a "homogeneous function of degree one" means?
$endgroup$
– Will M.
Nov 24 '18 at 22:43
$begingroup$
What does a "homogeneous function of degree one" means?
$endgroup$
– Will M.
Nov 24 '18 at 22:43
$begingroup$
A homogeneous function of degree 1 means that a function satisfies this property F(x.k)=k.F(x), k is a constant. A homogeneous function of degree n is F(x.k)=k^n . F(x)
$endgroup$
– Juan123
Nov 24 '18 at 22:58
$begingroup$
A homogeneous function of degree 1 means that a function satisfies this property F(x.k)=k.F(x), k is a constant. A homogeneous function of degree n is F(x.k)=k^n . F(x)
$endgroup$
– Juan123
Nov 24 '18 at 22:58
$begingroup$
Did you noticed then that $F(x)=xF(1)$ and so, $F$ is the function $y = kx$ with $k = F(1)$? Either you completely lost it, or else, I am not getting it.
$endgroup$
– Will M.
Nov 25 '18 at 2:34
$begingroup$
Did you noticed then that $F(x)=xF(1)$ and so, $F$ is the function $y = kx$ with $k = F(1)$? Either you completely lost it, or else, I am not getting it.
$endgroup$
– Will M.
Nov 25 '18 at 2:34
add a comment |
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$begingroup$
What does a "homogeneous function of degree one" means?
$endgroup$
– Will M.
Nov 24 '18 at 22:43
$begingroup$
A homogeneous function of degree 1 means that a function satisfies this property F(x.k)=k.F(x), k is a constant. A homogeneous function of degree n is F(x.k)=k^n . F(x)
$endgroup$
– Juan123
Nov 24 '18 at 22:58
$begingroup$
Did you noticed then that $F(x)=xF(1)$ and so, $F$ is the function $y = kx$ with $k = F(1)$? Either you completely lost it, or else, I am not getting it.
$endgroup$
– Will M.
Nov 25 '18 at 2:34