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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!

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    $\begingroup$ As @Xi'an points out in his answer, you probably mean $E[X_1+X_2]=0$, e.g. 1 and -1. $\endgroup$ – PaulG Feb 21 at 18:30
  • $\begingroup$ @PaulG Thank you very much. Could youi please have a look at this question as well. I have been thinking about this for long : stats.stackexchange.com/questions/510523/… $\endgroup$ – MathMan Feb 21 at 18:34
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If $X_1+\ldots+X_n=0$ with probability one, the vector $(X_1,\ldots,X_n)$ is not a Normal vector, hence $X^2_1+\ldots+X^2_n$ is not a $\chi^2_n$ random variable.

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