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Would someone please point me to a layman's reference, or walk me through the procedure that was used to determine that the variance of the logistic distribution is and the kurtosis is 6/5? The article on kurtosis, cumulant, and the like lost me, I'm afraid. I have a probability density function that's a nonlinear (but symmetrical) distortion of the logistic distribution, and I'd like to know its kurtosis and variance. If I know how to do it for the logistic distribution, I can figure it out for my version.
I can figure them out with a brute-force approach by generating thousands of values distributed according to my distribution, and then calculating the variance and kurtosis as . That will give me an approximation. I need help figuring it out exactly, not numerically.
According to the moment (mathematics) article, I would find, say, the variance of the logistic distribution like this:
...but when I plug the indefinite integral into into Wolfram Alpha (which serves as a Mathematica front end), I get a result that involves a polylogarithm, which isn't exactly closed form. Wolfram Alpha won't solve the definite integral, and by inspection, substituting the -∞ and +∞ for x won't give a meaningful result. So where does come from, as given in the logistic distribution article? ~ Amatulić ( talk) 04:52, 30 April 2010 (UTC)
Upon further investigation, it looks like the optimum value of a for my data is around 1.7.
Yes, my value of a is fixed once I determine the best fit to my data.
I was wondering if there was an elegant way to express in terms of a. I painstakingly solved for s=1 and a large number of values of a and, by inspection and experimentation, discovered five closed-form results in addition to the one you found (shown last):
Let
(Corrected per reply comments below)
Those are just the ones I found. There may be more. For all I know, the whole relationship between sigma, a, and s might be a closed-form expression. That's one thing I was curious about.
If you look at the plot, you see that looks like it should be capable of being modeled somehow. The discontinuity at a=1 is where my distribution reduces to something Cauchy-like. I have no explanation for the behavior at a<1; in fact, between 0 and 1/2 there is only one value, a=1/4, that has a solution for variance.
While it's true that kurtosis can't be less than -3, mathematically a definite integral can have any value it wants. My red plot of kurtosis shows that it isn't defined at a=2, but also shows a smooth curve of negative values where a<2. Nevertheless, your comment above suggests that this may be an artifact of the online definite integral solver I used. Wolfram Alpha doesn't give me answers for a<2.
As to having a choice of distributions... I made this one up because it meets all the following requirements where other distributions fail:
I couldn't find anything else that met all those requirements.
One thing I want to do with this is modify the Black-Scholes option pricing formula, but for that I need kurtosis. On the other hand, the adjustment doesn't blow up with kurtosis values < -3. I'll try it and see (assuming those negative values I got are real results). ~ Amatulić ( talk) 04:50, 2 May 2010 (UTC)
The plot at the right shows a histogram of my data overlayed by my distribution (a=1.6), a normal distribution, and a logistic distribution plotted in log scale. My distribution looks like a perfect fit when plotted in linear scale; I use the log scale because I am most interested in the tails.
One thing I'm still confused about is kurtosis. I have a distribution that fits my data better than anything else I have found, and my distribution has infinite kurtosis at a=1.6 which provides the best fit. And yet, I can calculate the sample kurtosis directly from my data to get a reasonable positive value of 29.6. You can see visually the data and my distribution are a good match. So why does one have infinite kurtosis and not the other? ~ Amatulić ( talk) 21:27, 4 May 2010 (UTC)
Hi, I'm trying to solve this GeoCaching puzzle, and it's driving me mad. I need to find: "the lowest 10 digit integer which contains each of the digits 0 to 9 and which is exactly divisible by every number from 2 to 16 inclusive". I've tried working it out by testing each combination in order - I've got up to 1237546890 with no joy so far - is there a quicker way to solve this? The only digit I.m certain of is the zero, which has to be last for the number to be divisible by 10. Can anyone help me please? —Preceding unsigned comment added by 194.205.143.136 ( talk) 05:42, 30 April 2010 (UTC)
(ec)
program mp ! Math Problem.
integer I,J,N,C(10),CHECK
character*11 STRING
do I=1713,13703
N = I*720720
write (STRING,*) N
STRING = STRING(2:) ! Needed since default format puts N
! in STRING(2:11),
! not STRING(1:10).
do J=1,10
C(J) = 0
enddo
do J=1,10 ! Count number of each digit:
! C(1) = number of ones in
! current number, etc.
if (STRING(J:J) .eq. "1") C( 1) = C( 1) + 1
if (STRING(J:J) .eq. "2") C( 2) = C( 2) + 1
if (STRING(J:J) .eq. "3") C( 3) = C( 3) + 1
if (STRING(J:J) .eq. "4") C( 4) = C( 4) + 1
if (STRING(J:J) .eq. "5") C( 5) = C( 5) + 1
if (STRING(J:J) .eq. "6") C( 6) = C( 6) + 1
if (STRING(J:J) .eq. "7") C( 7) = C( 7) + 1
if (STRING(J:J) .eq. "8") C( 8) = C( 8) + 1
if (STRING(J:J) .eq. "9") C( 9) = C( 9) + 1
if (STRING(J:J) .eq. "0") C(10) = C(10) + 1
enddo
CHECK = 0 ! Number of digits found
! exactly once in string.
do J=1,10
if (C(J) .eq. 1) CHECK = CHECK + 1
enddo
if (CHECK .eq. 10) print *,STRING
enddo
end
forstep(n=0,10^10,720720,if(vecsort(Col(Str(n)))==Col("0123456789"),print(n)))
program mp ! Math Problem.
integer*8 I,J,K,N,C(10),CHECK
character*11 STRING
K = 0 ! Number of solutions found.
do I=1421,13703
N = I*720720
write (STRING,*) N
STRING = STRING(2:) ! Needed since default format puts N
! in STRING(2:11),
! not STRING(1:10).
do J=1,10
C(J) = 0
enddo
do J=1,10 ! Count number of each digit:
! C(1) = number of ones in
! current number, etc.
if (STRING(J:J) .eq. "1") C( 1) = C( 1) + 1
if (STRING(J:J) .eq. "2") C( 2) = C( 2) + 1
if (STRING(J:J) .eq. "3") C( 3) = C( 3) + 1
if (STRING(J:J) .eq. "4") C( 4) = C( 4) + 1
if (STRING(J:J) .eq. "5") C( 5) = C( 5) + 1
if (STRING(J:J) .eq. "6") C( 6) = C( 6) + 1
if (STRING(J:J) .eq. "7") C( 7) = C( 7) + 1
if (STRING(J:J) .eq. "8") C( 8) = C( 8) + 1
if (STRING(J:J) .eq. "9") C( 9) = C( 9) + 1
if (STRING(J:J) .eq. "0") C(10) = C(10) + 1
enddo
CHECK = 0 ! Number of digits found
! exactly once in string.
do J=1,10
if (C(J) .eq. 1) CHECK = CHECK + 1
enddo
if (CHECK .eq. 10) THEN
K = K + 1 ! Increment number of solutions found.
print *,K," = ",STRING
endif
enddo
end
I enjoyed coding in the J (programming language) the lowest 10 digit integer which contains each of the digits 0 to 9 and which is exactly divisible by every number from 2 to 16 inclusive, and the number of solutions
({.,#) (#~(n=(\:])"_1&.((10#10)&#:))) (*[:i.[:<.(n=.9876543210x)&%) *./2+i.15 1274953680 66
I don't know any shortcut, but brute force works. Bo Jacoby ( talk) 13:00, 1 May 2010 (UTC).
Hi all - I'm working my way through some old exam papers and I came upon this quick problem which I can't for the life of me seem to solve: could anyone give me a hand?
If A is a n*n matrix and all entries of A are real and positive, do all eigenvalues of A have positive real part? I know that since all entries are real the determinant (polynomial) will have only real coefficients so for any complex eigenvalue root, its conjugate must be an eigenvalue too - then I tried looking at the trace as the sum of the eigenvalues, and arguing that summing over all the eigenvalues will end up picking up only the real parts of every eigenvalue, since every complex root has a conjugate eigenvalue, so the trace will be the sum of the real parts of the eigenvalues - and certainly, since all entries >= 0, the trace >= 0, but that only tells me about the overall sum of the eigenvalues being positive - for all I know, there could be a negative eigenvalue and just an even larger positive eigenvalue which makes the trace positive despite a negative eigenvalue, and I can't see any way to 'fix' the matrix so that we can definitely obtain only that negative-real-part eigenvalue without any of those larger positive-real-part eigenvalues necessarily being there: or who knows, perhaps it's in fact false and I simply can't think of a counterexample!
Your thoughts would be much appreciated, 131.111.29.210 ( talk) 15:18, 30 April 2010 (UTC)
Is there any any algorithm to describe the distribution of n points on a sphere such that the distance between points is maximized? For instance, if there were n entities of a certain length, with a charge (the same) at one end, and all joined at the other end, how would they orient themselves to minimize the potential energy? For certain cases, it's trivial (2 = opposite, 3 = triangle), and it should be nice and symmetric for the Platonic solids (4 = tetrahedron, 6 = cube, 8 = icosahedron, 12 = icosahedron, 20 = dodecahedron). For 5, I assume it would be a double tetrahedon, with three oriented along the horizontal plane in a triangle, and the other two directly up and down. Is there a general algorithm or pattern? Perhaps there is not a unique solution? — Knowledge Seeker দ 21:20, 30 April 2010 (UTC)
If you want to minimize the potential energy, then the problem is not trivial, see here for an analogous problem which is discussed here and the outline of a solution is given here. Count Iblis ( talk) 22:23, 30 April 2010 (UTC)
I think Coxeter's 12 Geometric Essays book addresses this problem. A Dover reprint bears the title The Beauty of Geometry, if I recall correctly. Michael Hardy ( talk) 20:54, 1 May 2010 (UTC)
What is the antiderivative of square root. 76.199.144.250 ( talk) 22:14, 30 April 2010 (UTC)
Let U, A and B be Banach spaces, L(A,B) the vector space of linear maps from A to B, let
I'm wandering whether some version of the product rule holds like
and in that case what would be the meaning of the product DM(x)v(x).
Can anybody help me?-- Pokipsy76 ( talk) 22:31, 30 April 2010 (UTC)
Mathematics desk | ||
---|---|---|
< April 29 | << Mar | April | May >> | May 1 > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
Would someone please point me to a layman's reference, or walk me through the procedure that was used to determine that the variance of the logistic distribution is and the kurtosis is 6/5? The article on kurtosis, cumulant, and the like lost me, I'm afraid. I have a probability density function that's a nonlinear (but symmetrical) distortion of the logistic distribution, and I'd like to know its kurtosis and variance. If I know how to do it for the logistic distribution, I can figure it out for my version.
I can figure them out with a brute-force approach by generating thousands of values distributed according to my distribution, and then calculating the variance and kurtosis as . That will give me an approximation. I need help figuring it out exactly, not numerically.
According to the moment (mathematics) article, I would find, say, the variance of the logistic distribution like this:
...but when I plug the indefinite integral into into Wolfram Alpha (which serves as a Mathematica front end), I get a result that involves a polylogarithm, which isn't exactly closed form. Wolfram Alpha won't solve the definite integral, and by inspection, substituting the -∞ and +∞ for x won't give a meaningful result. So where does come from, as given in the logistic distribution article? ~ Amatulić ( talk) 04:52, 30 April 2010 (UTC)
Upon further investigation, it looks like the optimum value of a for my data is around 1.7.
Yes, my value of a is fixed once I determine the best fit to my data.
I was wondering if there was an elegant way to express in terms of a. I painstakingly solved for s=1 and a large number of values of a and, by inspection and experimentation, discovered five closed-form results in addition to the one you found (shown last):
Let
(Corrected per reply comments below)
Those are just the ones I found. There may be more. For all I know, the whole relationship between sigma, a, and s might be a closed-form expression. That's one thing I was curious about.
If you look at the plot, you see that looks like it should be capable of being modeled somehow. The discontinuity at a=1 is where my distribution reduces to something Cauchy-like. I have no explanation for the behavior at a<1; in fact, between 0 and 1/2 there is only one value, a=1/4, that has a solution for variance.
While it's true that kurtosis can't be less than -3, mathematically a definite integral can have any value it wants. My red plot of kurtosis shows that it isn't defined at a=2, but also shows a smooth curve of negative values where a<2. Nevertheless, your comment above suggests that this may be an artifact of the online definite integral solver I used. Wolfram Alpha doesn't give me answers for a<2.
As to having a choice of distributions... I made this one up because it meets all the following requirements where other distributions fail:
I couldn't find anything else that met all those requirements.
One thing I want to do with this is modify the Black-Scholes option pricing formula, but for that I need kurtosis. On the other hand, the adjustment doesn't blow up with kurtosis values < -3. I'll try it and see (assuming those negative values I got are real results). ~ Amatulić ( talk) 04:50, 2 May 2010 (UTC)
The plot at the right shows a histogram of my data overlayed by my distribution (a=1.6), a normal distribution, and a logistic distribution plotted in log scale. My distribution looks like a perfect fit when plotted in linear scale; I use the log scale because I am most interested in the tails.
One thing I'm still confused about is kurtosis. I have a distribution that fits my data better than anything else I have found, and my distribution has infinite kurtosis at a=1.6 which provides the best fit. And yet, I can calculate the sample kurtosis directly from my data to get a reasonable positive value of 29.6. You can see visually the data and my distribution are a good match. So why does one have infinite kurtosis and not the other? ~ Amatulić ( talk) 21:27, 4 May 2010 (UTC)
Hi, I'm trying to solve this GeoCaching puzzle, and it's driving me mad. I need to find: "the lowest 10 digit integer which contains each of the digits 0 to 9 and which is exactly divisible by every number from 2 to 16 inclusive". I've tried working it out by testing each combination in order - I've got up to 1237546890 with no joy so far - is there a quicker way to solve this? The only digit I.m certain of is the zero, which has to be last for the number to be divisible by 10. Can anyone help me please? —Preceding unsigned comment added by 194.205.143.136 ( talk) 05:42, 30 April 2010 (UTC)
(ec)
program mp ! Math Problem.
integer I,J,N,C(10),CHECK
character*11 STRING
do I=1713,13703
N = I*720720
write (STRING,*) N
STRING = STRING(2:) ! Needed since default format puts N
! in STRING(2:11),
! not STRING(1:10).
do J=1,10
C(J) = 0
enddo
do J=1,10 ! Count number of each digit:
! C(1) = number of ones in
! current number, etc.
if (STRING(J:J) .eq. "1") C( 1) = C( 1) + 1
if (STRING(J:J) .eq. "2") C( 2) = C( 2) + 1
if (STRING(J:J) .eq. "3") C( 3) = C( 3) + 1
if (STRING(J:J) .eq. "4") C( 4) = C( 4) + 1
if (STRING(J:J) .eq. "5") C( 5) = C( 5) + 1
if (STRING(J:J) .eq. "6") C( 6) = C( 6) + 1
if (STRING(J:J) .eq. "7") C( 7) = C( 7) + 1
if (STRING(J:J) .eq. "8") C( 8) = C( 8) + 1
if (STRING(J:J) .eq. "9") C( 9) = C( 9) + 1
if (STRING(J:J) .eq. "0") C(10) = C(10) + 1
enddo
CHECK = 0 ! Number of digits found
! exactly once in string.
do J=1,10
if (C(J) .eq. 1) CHECK = CHECK + 1
enddo
if (CHECK .eq. 10) print *,STRING
enddo
end
forstep(n=0,10^10,720720,if(vecsort(Col(Str(n)))==Col("0123456789"),print(n)))
program mp ! Math Problem.
integer*8 I,J,K,N,C(10),CHECK
character*11 STRING
K = 0 ! Number of solutions found.
do I=1421,13703
N = I*720720
write (STRING,*) N
STRING = STRING(2:) ! Needed since default format puts N
! in STRING(2:11),
! not STRING(1:10).
do J=1,10
C(J) = 0
enddo
do J=1,10 ! Count number of each digit:
! C(1) = number of ones in
! current number, etc.
if (STRING(J:J) .eq. "1") C( 1) = C( 1) + 1
if (STRING(J:J) .eq. "2") C( 2) = C( 2) + 1
if (STRING(J:J) .eq. "3") C( 3) = C( 3) + 1
if (STRING(J:J) .eq. "4") C( 4) = C( 4) + 1
if (STRING(J:J) .eq. "5") C( 5) = C( 5) + 1
if (STRING(J:J) .eq. "6") C( 6) = C( 6) + 1
if (STRING(J:J) .eq. "7") C( 7) = C( 7) + 1
if (STRING(J:J) .eq. "8") C( 8) = C( 8) + 1
if (STRING(J:J) .eq. "9") C( 9) = C( 9) + 1
if (STRING(J:J) .eq. "0") C(10) = C(10) + 1
enddo
CHECK = 0 ! Number of digits found
! exactly once in string.
do J=1,10
if (C(J) .eq. 1) CHECK = CHECK + 1
enddo
if (CHECK .eq. 10) THEN
K = K + 1 ! Increment number of solutions found.
print *,K," = ",STRING
endif
enddo
end
I enjoyed coding in the J (programming language) the lowest 10 digit integer which contains each of the digits 0 to 9 and which is exactly divisible by every number from 2 to 16 inclusive, and the number of solutions
({.,#) (#~(n=(\:])"_1&.((10#10)&#:))) (*[:i.[:<.(n=.9876543210x)&%) *./2+i.15 1274953680 66
I don't know any shortcut, but brute force works. Bo Jacoby ( talk) 13:00, 1 May 2010 (UTC).
Hi all - I'm working my way through some old exam papers and I came upon this quick problem which I can't for the life of me seem to solve: could anyone give me a hand?
If A is a n*n matrix and all entries of A are real and positive, do all eigenvalues of A have positive real part? I know that since all entries are real the determinant (polynomial) will have only real coefficients so for any complex eigenvalue root, its conjugate must be an eigenvalue too - then I tried looking at the trace as the sum of the eigenvalues, and arguing that summing over all the eigenvalues will end up picking up only the real parts of every eigenvalue, since every complex root has a conjugate eigenvalue, so the trace will be the sum of the real parts of the eigenvalues - and certainly, since all entries >= 0, the trace >= 0, but that only tells me about the overall sum of the eigenvalues being positive - for all I know, there could be a negative eigenvalue and just an even larger positive eigenvalue which makes the trace positive despite a negative eigenvalue, and I can't see any way to 'fix' the matrix so that we can definitely obtain only that negative-real-part eigenvalue without any of those larger positive-real-part eigenvalues necessarily being there: or who knows, perhaps it's in fact false and I simply can't think of a counterexample!
Your thoughts would be much appreciated, 131.111.29.210 ( talk) 15:18, 30 April 2010 (UTC)
Is there any any algorithm to describe the distribution of n points on a sphere such that the distance between points is maximized? For instance, if there were n entities of a certain length, with a charge (the same) at one end, and all joined at the other end, how would they orient themselves to minimize the potential energy? For certain cases, it's trivial (2 = opposite, 3 = triangle), and it should be nice and symmetric for the Platonic solids (4 = tetrahedron, 6 = cube, 8 = icosahedron, 12 = icosahedron, 20 = dodecahedron). For 5, I assume it would be a double tetrahedon, with three oriented along the horizontal plane in a triangle, and the other two directly up and down. Is there a general algorithm or pattern? Perhaps there is not a unique solution? — Knowledge Seeker দ 21:20, 30 April 2010 (UTC)
If you want to minimize the potential energy, then the problem is not trivial, see here for an analogous problem which is discussed here and the outline of a solution is given here. Count Iblis ( talk) 22:23, 30 April 2010 (UTC)
I think Coxeter's 12 Geometric Essays book addresses this problem. A Dover reprint bears the title The Beauty of Geometry, if I recall correctly. Michael Hardy ( talk) 20:54, 1 May 2010 (UTC)
What is the antiderivative of square root. 76.199.144.250 ( talk) 22:14, 30 April 2010 (UTC)
Let U, A and B be Banach spaces, L(A,B) the vector space of linear maps from A to B, let
I'm wandering whether some version of the product rule holds like
and in that case what would be the meaning of the product DM(x)v(x).
Can anybody help me?-- Pokipsy76 ( talk) 22:31, 30 April 2010 (UTC)