- #1
Li(n)
- 13
- 0
By the way , the Z^n part is supposed to be lowered case , sorry.
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Can you prove that Re{n} > -1/2 without using a prefix?
- Context: Graduate
- Thread starter Li(n)
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Discussion Overview
The discussion revolves around proving that Re{n} > -1/2 without using a prefix. Participants explore various mathematical approaches and concepts related to this assertion, including transformations and distributions.
Discussion Character
- Exploratory
- Technical explanation
- Mathematical reasoning
Main Points Raised
- Some participants suggest using a change of variable, specifically t = cos θ, as a potential method in the proof.
- Others note that the second last inequality may involve the moment generating function (MGF) for the chi-squared distribution and mention the relationship between chi-squared distributions and sums of squared standard normal variables.
- A participant recalls the gamma function's role in the symmetry of geometric Brownian motion, proposing a possible connection to the current problem.
- One participant references a Wikipedia article on chi-squared distributions, indicating that changing to polar coordinates might be relevant to the proof.
- Another participant expresses uncertainty about complex transforms and questions whether De Moivre's theorem might be applicable in this context.
Areas of Agreement / Disagreement
Participants do not appear to reach a consensus, as multiple approaches and ideas are presented without resolution or agreement on a specific method to prove the assertion.
Contextual Notes
Some assumptions and dependencies on definitions are not fully explored, particularly regarding the use of complex transforms and the implications of the chi-squared distribution in the context of the proof.
By the way , the Z^n part is supposed to be lowered case , sorry.
- #2
Li(n)
- 13
- 0
What if I Use the change of variable t = cos \theta
- #3
Li(n)
- 13
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Here are some thoughts : At a glance, the second last inequality contains the MGF for the chi-squared distribution, and just looking at the integrals, the change of coordinates for the normal distribution may be involved somewhere in there. Consider also that the chi-square distribution with n - 1 degrees of freedom is the limit distribution of a sum of n Z^2 random variables, where Z is the standard normal.
I also recall seeing the gamma function in the proof of the symmetry of geometric brownian motion about the x axis, so that may be distantly related.
I also recall seeing the gamma function in the proof of the symmetry of geometric brownian motion about the x axis, so that may be distantly related.
- #4
Li(n)
- 13
- 0
http://en.wikipedia.org/wiki/Proofs_related_to_chi-squared_distribution
I found something , "derivation of the pdf for k degrees of freedom":
The rest is just a matter of changing to polar coordinates.
I'm not too well-versed with complex transforms, though, since there's a complex number. I think this is a clue whether I'm on the right track or not if something like De Moivre's theorem fits in very nicely when changing to polar coordinates.
I found something , "derivation of the pdf for k degrees of freedom":
The rest is just a matter of changing to polar coordinates.
I'm not too well-versed with complex transforms, though, since there's a complex number. I think this is a clue whether I'm on the right track or not if something like De Moivre's theorem fits in very nicely when changing to polar coordinates.
- #5
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What is your question?
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