Combining non-independent random normal-distributed variables

Question

In the below question part b) involves combining normally-distributed random variables which ARE independent.

Part d) involves combining normally-distributed random variables which are NOT independent.

I understand part b) but what is the general approach to solve part d)? Re-write the equation without referring to the sample mean of U?

Question:

Answer 1

Re-write the equation without referring to the sample mean of U?

That's a good idea. $U_1 - \bar{U} = \frac{4}{5} U_1 - \frac{1}{5} (U_2 + \cdots + U_5)$. This is a normal random variable with mean $0$ and variance $\frac{4^2 + 4}{5^2} \sigma^2 = \frac{4}{5}\sigma^2$. Can you then compute $P(U_1 - \bar{U} > \sigma)$?