I understand that
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I do not understand why
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Thanks! So basically, whats the rule with chi-squared distributions and degrees of freedom. My book is getting very frustrating because they're starting to use certain concepts a little to liberally without explaining where they came from or why they are in the form the are; this is a good example.
Thanks again.
follows a normal distribution.
so follows a chi-square distribution with 1 degree of freedom
and follows a chi-square distribution with n degrees of freedom (since you are summing n independent chi-square variables)
so you would expect that to follow a chi-square distribution with n-2 degrees of freedom because you lose two degrees of freedom estimating the two regression coefficients
I don't agree with this...........
I agree with ....
follows a chi-square distribution with 1 degree of freedom
and
and follows a chi-square distribution with n-1 degrees of freedom
while
and follows a chi-square distribution with n degrees of freedom (since you are summing n independent chi-square variables)
In response to matheagle, let me change a few things:
follows a normal distribution.
so follows a chi-square distribution with 1 degree of freedom
and follows a chi-square distribution with n degrees of freedom (since you are summing n independent chi-square variables)
so you would expect that to follow a chi-square distribution with n-2 degrees of freedom because you lose two degrees of freedom estimating the two regression coefficients
Got it.
Sure.
What? How do you lose degrees of freedom? I think I'm lacking a fundamental understanding of just what a degree of freedom is. I understand how it relates to the Gamma distribution and it is actually the mean of the Chi-square distribution. But I wouldn't know something followed a Chi-square distribution if it was staring me in the face, let alone how many degrees of freedom it employs.
Degrees of freedom can be looked at as the number of independent pieces of information that go into estimating a parameter. By first estimating the 2 regression coefficients, you are constraining the residuals to lie within a certain vector space defined by the 2 normal equations. Therefore, you are working with two less independent pieces of information.