I am having some troubles with a question regarding a joint distribution with several normal distributions.
The question is put up like this:
$X_1 , X_2$ are two normal distributions with the distribution $N(0,I_1) $. $Z_1 = \frac{X_1 +X_2}{2}, Z_2 = \frac{X_1 -X_2}{2}, Z_1 = \frac{X_2 -X_1}{2}$.
Find the joint distribution of $(Z_1, Z_2, Z_3)$.
What I did so far was to try find the distributions of $Z_1, Z_2, Z_3$, where I used the rules for sums of normal distributions. This gives me the result:
$Z_1 \sim N(\frac{1}{2}\cdot 0 + \frac{1}{2}\cdot 0, \frac{1}{2}^2\cdot I_1 +\frac{1}{2}^2\cdot I_1 ) = N(0, \frac{1}{2}I_1)$ $Z_2 \sim N(\frac{1}{2}\cdot 0 + \frac{-1}{2}\cdot 0, \frac{1}{2}^2\cdot I_1 +\frac{-1}{2}^2\cdot I_1 ) = N(0, \frac{1}{2}I_1)$ $Z_2 \sim N(\frac{1}{2}\cdot 0 + \frac{-1}{2}\cdot 0, \frac{1}{2}^2\cdot I_1 +\frac{-1}{2}^2\cdot I_1 ) = N(0, \frac{1}{2}I_1)$
I then try using the rules for multivariate normal distributions to find the joint distribution of $Z_1, Z_2, Z_3 $.
This gives: $ Z \sim N(\mu , \sum) $
where
$\mu = [E[Z_1],E[Z_2], E[Z_3]]$
$\sum = [Cov[Z_i,Z_j]] \; i = 1,2,3 \; j = 1,2,3$
This however just gives me a vector with zeroes and a matrix with zeroes with my calculations. What am I doing wrong in my assumptions?