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I am trying to understand the degrees of freedom parameter in the Chi_squared distribution and I have found two references from the same source that appear, to me, to contradict one-another. Can anyone explain what is going on ?

In https://onlinecourses.science.psu.edu/stat414/node/171 it states that:

Corollary. If X1, X2 , ... , Xn are independent normal random variables with mean=0 and variance=1, that is: Xi∼N(0,1) for i = 1, 2, ..., n. Then:

Sum_from_one_to_n(Xi)^2 ∼ χ2(n) ( I have simplified the formula in the source by setting all μi to 0 and σi to 1)

But https://onlinecourses.science.psu.edu/stat414/node/174 states that:

X1, X2, ... , Xn are observations of a random sample of size n from the normal distribution N(0,1) then:

Sum_from_one_to_n(Xi)^2 ∼ χ2(n-1) (Again setting all μi to 0 and σi to 1)

So it seems that a (slightly) different pdf is being given for the same R.V. , we have lost 1 degree of freedom. Can anyone explain this, or does the fact that the individual observations in the second case are described as a 'random sample' somehow impact on their independence ? If so, how exactly, is each element of the sample not an RV in its own right whose pdf is that of the population and => N(0,1) ?

Thanks,

Usjes

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# How many degrees of freedom ?

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