# Are Chi Squared Random Variables Independent from Standard Normal Random Variables?

My intuition on this says, no, since the $$\chi^2$$ random variable $$U=\sum_{i=1}^k Z_i^2$$ where each $$Z_i$$ is an independent Standard Normal random variable.

That said, I was told by a professor some time ago that the sample mean is independent from the sample variance of a Normal distribution, so does it follow that U is independent from Z?

• First think about this: In $U=Z^2$ (i.e. $k=1$), is $U$ independent of $Z$? Aug 4, 2019 at 14:41

Your professor's comment is about Basu's theorem, see the Example section also. As for your question, which is also counter-intuitive, think about the case where $$k=1$$, i.e. $$U=Z^2$$. A simple proof of dependence would be to show a non-zero covariance value, however these variables have zero covariance. Another one is to think of conditional distribution $$f_{U|Z}(u|z)$$. Independence forces us to have $$f_{U|Z}(u|z)=f_U(u)$$ which is not correct since $$f_{U|Z}(u|z)$$ have a degenerate PDF which is equal to $$\delta(u-z^2)\neq f_U(u)$$, not a Chi-Squared distribution. Intuitively, if we know $$Z=z$$, then $$U$$ will certainly be $$u=z^2$$ without any uncertainty left.