I am new to Statistics and trying to intuitively understand how a change in degrees of freedom affects the mean of a chi-square distribution.
Suppose, We have $n$ normal random variables such that $X_1 + \cdots + X_n = 0$
Now, for a chi-square distribution, the expected value is = degrees of freedom.
In this case, the degrees of freedom is $n-1$.
If we consider an example where $X_1 + X_2 = 0 $.
Then $X_1^2 + X_2^2 = 2 X_1^2$ although has $1$ degree of freedom yet it's mean value is $2$!
Where am I making a mistake?
Thanks a lot!