Given independent Gaussian random variables $U\sim N(−1,1)$ and $V\sim N(1,1)$, are the 2-element vector $T=(U+V, U−2V)$ and the variable $$W= U\text{ with 50% chance}, V \text{ with 50% chance}$$ also Gaussian? If yes, what is the mean and variance (covariance matrix)?
For simple versions of Gaussian problems like $U+V$ I can easily identify properties, but how to approach forms as above? Like for the vector one, what is the condition for it to be Gaussian random var anyway? And how probability of $U$ and $V$ will affect the identity?
Any suggestions?