Let $Y_1, Y_2, Y_3, Y_4 \sim N(\beta_0, \sigma^2)$, $Y_i$'s independent.
Given the vector $Y_{4 \times 1} = [Y_1, Y_2, Y_3, Y_4]^T$ and the orthonormal
coordinate system $U_1 = \frac{1}{2}[1, 1, 1, 1]^T$, $U_2 = \frac{1}{2}[1, 1, -1, -1]^T$,
$U_3 = \frac{1}{2}[1, -1, 1, -1]^T$, and $U_4 = \frac{1}{2}[1, -1, -1, 1]^T$.
Find $c_1, c_2, c_3, c_4$ for which $Y = c_1U_1 + c_2U_2 + c_3U_3 + c_4U_4$.
The distribution of $c_1$ is
Chi-square
F distribution
T distribution
Normal
