How would you go about proving that
[ ( n - 1 ) * s^2 ] / σ^2
has a Chi-Square Distribution?
Hello,
Some information necessary about s^2
Anyway, I can just guess what it is. Note that aX~N(0,a²), where X~N(0,1)
Now think about what happens if you divide a normal distribution N(0,σ²) by σ² !
And finally, remember that a chi-square distribution is the sum of squares of iid random variables following a N(0,1)
One way to get at this is to assume , prove it for the case n = 2, then induct by showing that
,
which will get you where you need since you will have the sum of two independent chi-squares. Some facts that you will use are that the sum of independent chi-squares is chi-square, that the square of an is , and that . The definitions are:
Once you have this, it's just a matter of undoing the assumption that , which isn't really a big deal. To be honest, if this is just for kicks, it's actually more of a pain in the head than it appears, particularly in proving the identity above.