- #1
Li(n)
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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?
- Thread starter Li(n)
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In summary, the conversation discusses the use of change of variables in calculations involving the chi-squared distribution and the normal distribution. The relationship between the chi-squared distribution and a sum of n random variables is mentioned, as well as the use of polar coordinates. The speaker also mentions their lack of familiarity with complex transforms.
By the way , the Z^n part is supposed to be lowered case , sorry.
- #2
Li(n)
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What if I Use the change of variable t = cos \theta
- #3
Li(n)
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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)
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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?
1. What does "Re{n} > -1/2" mean?
The statement "Re{n} > -1/2" is a mathematical expression that represents the real part of a complex number, denoted by Re{n}, being greater than -1/2. In other words, the real part of the complex number is a positive value that is larger than -1/2.
2. Why is it important to prove that "Re{n} > -1/2"?
Proving that "Re{n} > -1/2" is important in many areas of mathematics and science, including complex analysis, differential equations, and physics. It is used to determine the stability of solutions to differential equations and to analyze the behavior of systems in physics and engineering.
3. How can one prove that "Re{n} > -1/2"?
There are several methods that can be used to prove that "Re{n} > -1/2". One approach is to use the definition of a complex number and show that the real part is greater than -1/2. Another method is to use mathematical properties and theorems, such as the triangle inequality, to manipulate the expression and arrive at the desired result.
4. Can "Re{n} > -1/2" be proven for all values of n?
Yes, "Re{n} > -1/2" can be proven for all values of n. This statement is a general rule that applies to all complex numbers, regardless of their specific values. Therefore, the proof holds for all possible values of n.
5. What are the implications of "Re{n} > -1/2" in practical applications?
The statement "Re{n} > -1/2" has many practical implications, such as identifying stable and unstable solutions in differential equations, analyzing the behavior of electronic circuits, and predicting the stability of physical systems. It is an important concept that is used in various fields, including engineering, physics, and mathematics.
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