[math-fun] names for some well known facts
in probability theory: variance + mean^2 = meansquare in minimization: Let F(x1,x2,x3,...,xN) be a concave-U symmetric function of N variables. Then its min necessarily occurs when all the variables are equal. The above are two well known theorems. What I want to know is, do they have names, or cites, or something like that so I do not have to re-prove them myself, I can just name/cite them? There was once a sci-fi book with title "True names" postulating that once I know your true name, I have great power...
in probability theory: variance + mean^2 = meansquare
This is the one-dimensional special case of the Huygens-Steiner theorem, which states: |X - E(X)|^2 + |E(X) - y|^2 = E(|X - y|^2) where X is a random variable on an inner product space and y is an arbitrary point in the inner product space. A generalisation is that: |X - E(X)|^2 + |Y - E(Y)^2| + |E(X) - E(Y)|^2 = E(|X - Y|^2) where X and Y are two independent random variables on the same inner product space. Equivalently: "The mean squared distance between two distributions is equal to the sum of their variances and the squared distance between their barycentres." This can be proved by two applications of Huygens-Steiner. Best wishes, Adam P. Goucher
Maybe the second term should have the ^2 outside the || ? —Dan
On Feb 22, 2016, at 3:58 AM, Adam P. Goucher <apgoucher@gmx.com> wrote:
A generalisation is that:
|X - E(X)|^2 + |Y - E(Y)^2| + |E(X) - E(Y)|^2 = E(|X - Y|^2)
where X and Y are two independent random variables on the same inner product space. Equivalently:
"The mean squared distance between two distributions is equal to the sum of their variances and the squared distance between their barycentres."
Oh, yes, well spotted... -- APG.
Sent: Monday, February 22, 2016 at 5:25 PM From: "Dan Asimov" <asimov@msri.org> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] names for some well known facts
Maybe the second term should have the ^2 outside the || ?
—Dan
On Feb 22, 2016, at 3:58 AM, Adam P. Goucher <apgoucher@gmx.com> wrote:
A generalisation is that:
|X - E(X)|^2 + |Y - E(Y)^2| + |E(X) - E(Y)|^2 = E(|X - Y|^2)
where X and Y are two independent random variables on the same inner product space. Equivalently:
"The mean squared distance between two distributions is equal to the sum of their variances and the squared distance between their barycentres."
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Your odds are pretty good if you attribute results to Euler.
On Feb 19, 2016, at 11:47 PM, Warren D Smith <warren.wds@gmail.com> wrote:
in probability theory: variance + mean^2 = meansquare
in minimization: Let F(x1,x2,x3,...,xN) be a concave-U symmetric function of N variables. Then its min necessarily occurs when all the variables are equal.
The above are two well known theorems. What I want to know is, do they have names, or cites, or something like that so I do not have to re-prove them myself, I can just name/cite them?
There was once a sci-fi book with title "True names" postulating that once I know your true name, I have great power...
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Or Gauss. Or Newton. WFL On 2/22/16, Veit Elser <ve10@cornell.edu> wrote:
Your odds are pretty good if you attribute results to Euler.
On Feb 19, 2016, at 11:47 PM, Warren D Smith <warren.wds@gmail.com> wrote:
in probability theory: variance + mean^2 = meansquare
in minimization: Let F(x1,x2,x3,...,xN) be a concave-U symmetric function of N variables. Then its min necessarily occurs when all the variables are equal.
The above are two well known theorems. What I want to know is, do they have names, or cites, or something like that so I do not have to re-prove them myself, I can just name/cite them?
There was once a sci-fi book with title "True names" postulating that once I know your true name, I have great power...
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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participants (5)
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Adam P. Goucher -
Dan Asimov -
Fred Lunnon -
Veit Elser -
Warren D Smith