Determine independence/dependence of random variables

Question

Let $Z_1$ and $Z_2$ be independent standard normal random variables.

Let $W = \frac{Z_1 + Z_2}{\sqrt{2}}$ so that $W \sim N(0,1)$.

Let $U = Z_1^2 + Z_2^2$ so that $U \sim \chi_2^2$.

How can I determine if W and U are independent or dependent?

Answer 1

To prove the independence and dependence of those. You may use the moment generating function (MGF).

Hints: in page 10, of this document: https://ocw.mit.edu/courses/mathematics/18-443-statistics-for-applications-spring-2015/lecture-notes/MIT18_443S15_LEC1.pdf