Implication of relationship between multivariate normal distribution and chi-square distribution 
I am wondering what is the implication of the above relation/theorem. I know how to prove this using "sphering $Y$" but I am failing to get intuitive understanding of the theorem. What does it mean for $(Y-\mu)'\Sigma^{-1}(Y-\mu)$ to be distributed as $\chi^{2}_{n}$ ? What is the implication?
 A: If you by "intuitive understanding" mean how you can see this result instantly i your mind's eye:

*

*subtracting $\mu$ centers to zero mean


*rewrite the quadratic form to $ \left[\Sigma^{-1/2}(Y-\mu)\right]^T \left[\Sigma^{-1/2}(Y-\mu)\right]$. Calculate the covariance matrix of the bracketed term, you will find it is the identity matrix.


*This shows that the quadratic form has **the same distribution as the sum of squares of $n$ iid standard normal random variables.
It is not clear what you mean by asking

I am wondering what is the implication of the above relation/theorem

The only implication is that you now know the distribution of the quadratic form, and that might be useful.
A: What it says to me is that every chi square distribution can be thought of as describing the variance of a symmetric multivariate normal distribution (smnd). Which might be a very easy visualization of the kind of question the chi square is asking. This would relate to https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Interval , https://en.wikipedia.org/wiki/Chi-squared_distribution#History and the last 2 sentences (scroll up) above
https://en.wikipedia.org/wiki/Chi-squared_distribution#Probability_density_function
So a chi square test is asking how well your data corresponds to a multivariate normal distribution of appropriate dimensionality (which you specify using degrees of freedom) this measures whether your data is likely to arise randomly based on multinomial assumptions. agree?
