If $X_{1}, X_{2}, \ldots, X_{N}$ are identically distributed normal random variables with mean $0$ and variance $\frac{(N+3)D\sigma^{2}}{N}$, then I want to calculate the distribution, or at least the variance of $Z = X_{1}^{2} + X_{2}^{2} + \ldots + X_{N}^{2}$.

I already found that $Cov(X_{i}, X_{j}) = \frac{\sigma^{2}D}{N}$, $\forall i,j \in \{1, 2, \ldots, N\}$ and that $E[X_{i}^{2}] = \frac{(N+3)D\sigma^{2}}{N}$, $Var(X_{i}^{2}) = \frac{(N+3)^{2}{D^{2}\sigma}^{4}}{N^{2}}$, $\forall i,j \in \{1, 2, \ldots, N\}$.

My problem is that they are correlated and I do not know how to compute the variance and the distribution of $Z$.

Any help would be highly considered!

  • $\begingroup$ Rotate to new coordinates which are uncorrelated. $\endgroup$ – kimchi lover Apr 13 at 17:11
  • $\begingroup$ Thanks! How do I do that? $\endgroup$ – CLBJ_23 Apr 13 at 20:25
  • $\begingroup$ Your covariance matrix is a multiple of the diagonal matrix plus a rank 1 matrix, so it has $(N-1)$ eigenvalues that are equal plus one more, corresponding to the eigenvector of all $1$s. From which you can deduce that $Z$ is the sum of a multiple of a degree-of-freedom $N-1$ chi squared rv plus a different multiple of a degree-of-freedom $1$ chi square. $\endgroup$ – kimchi lover Apr 13 at 22:32

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