Division of Chi Squared Random Variables

[jojay99]

Hey guys,

I have a quick question. Suppose X is a chi squared random variable with n degrees of freedom and Y is another independent chi squared random variable with n degrees of freedom.

Is X/Y ~ 1 ?

Intuitively, it makes sense to me but I'm not too sure.

 

[chiro]

Hey jojay99 and welcome to the forums.

If X != Y the answer is an emphatic no. You can't just assume that X = Y just because both have the same degree of freedom: if they are two distinct random variables then they will have some distribution.

The result you should look at is the derivation of the F-distribution:

http://en.wikipedia.org/wiki/F-distribution

Any derivation of the F-distribution will tell you how the distribution for a ratio of chi-square distributions (with terms for the degrees which are constants) is derived.

Remember that if you have two distributions you need to check whether the two variables correspond to the same process and not the same variable definition or PDF.

Another thing to think about: is X + Y = 2X? How about X + Y = 2Y? Even if it's not a random variable, just think a normal variable and consider those questions.