Help with chi square distribution

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Discussion Overview

The discussion revolves around demonstrating that the sum of two independent chi-square distributed random variables, one with n degrees of freedom and the other with m degrees of freedom, results in a chi-square distribution with n+m degrees of freedom. Participants explore the use of moment generating functions (MGFs) to support this assertion.

Discussion Character

  • Technical explanation, Mathematical reasoning, Debate/contested

Main Points Raised

  • One participant asks how to show that the sum of two chi-square distributions results in another chi-square distribution using moment generating functions.
  • Another participant agrees with the initial question but does not provide additional information.
  • A third participant attempts to provide a solution by stating the moment generating functions for the two chi-square distributions and concludes that their product yields the MGF of a chi-square distribution with (n+m) degrees of freedom.
  • A fourth participant reiterates the original question and suggests starting with the Gaussian distribution to derive the characteristic function from the MGF to demonstrate the additivity of chi-square distributions.

Areas of Agreement / Disagreement

There is no consensus on the approach to demonstrate the claim, as participants present different methods and some simply restate the question without providing resolution.

Contextual Notes

The discussion does not clarify assumptions regarding the independence of the random variables or the specific conditions under which the moment generating functions are applicable.

sneaky666
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How do i show that the a [X1 has a chi square distribution with n degrees of freedom] + [X2 has a chi square distribution with m degrees of freedom] is a [X1+X2 has a chi square distribution with n+m degrees of freedom]?

How can i use moment generating functions to do this?
 
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Yes.
 
sneaky666 said:
How do i show that the a [X1 has a chi square distribution with n degrees of freedom] + [X2 has a chi square distribution with m degrees of freedom] is a [X1+X2 has a chi square distribution with n+m degrees of freedom]?

How can i use moment generating functions to do this?

MX1=(1-2t)-n/2
MX2=(1-2t)-m/2

MX1+X2=MX1*MX1= (1-2t)-(n+m)/2

which is the MGF of a chi-square distribution with (n+m) degrees of freedom.

That's all I can come up with...not terribly good with this...
 
sneaky666 said:
How do i show that the a [X1 has a chi square distribution with n degrees of freedom] + [X2 has a chi square distribution with m degrees of freedom] is a [X1+X2 has a chi square distribution with n+m degrees of freedom]?

How can i use moment generating functions to do this?

If you want to show that a given set of random variables has a chi square distribution and that these distributions are additive you need to start with the Gaussian and then derive the characteristic function from the MGF.

http://www.planetmathematics.com/CentralChiDistr.pdf
 

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