# Intuitive way to see how degrees of freedom affects the mean of a chi square distribution?

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!

• As @Xi'an points out in his answer, you probably mean $E[X_1+X_2]=0$, e.g. 1 and -1. Feb 21, 2021 at 18:30
• @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/… Feb 21, 2021 at 18:34

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.