Help with chi square distribution

[sneaky666]

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?

 

[statdad]

Yes.

 

[ych22]

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?

M_X1=(1-2t)^-n/2
M_X2=(1-2t)^-m/2

M_X1+X2=M_X1*M_X1= (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...

 

[SW VandeCarr]

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