Question

Let $Y_1, Y_2, Y_3, Y_4 \sim N(\beta_0, \sigma^2)$, $Y_i$'s independent. Given the vector $Y_{4\times1} = [Y_1, Y_2, Y_3, Y_4]^T$ 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 $U_4 = \frac{1}{2}[1, -1, -1, 1]^T$. First, 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 $\frac{c_2^2 + c_3^2 + c_4^2}{\sigma^2}$ is F distribution T distribution Chi-square Normal

          Let $Y_1, Y_2, Y_3, Y_4 \sim N(\beta_0, \sigma^2)$, $Y_i$'s independent.
Given the vector $Y_{4\times1} = [Y_1, Y_2, Y_3, Y_4]^T$ 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 $U_4 = \frac{1}{2}[1, -1, -1, 1]^T$.
First, 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 $\frac{c_2^2 + c_3^2 + c_4^2}{\sigma^2}$ is
F distribution
T distribution
Chi-square
Normal

Let Y1, Y2, Y3, Y4 ∼ N(β0, σ^2), Yi's independent.
Given the vector Y4×1 = [Y1, Y2, Y3, Y4]^T and the orthonormal
coordinate system U1 = (1)/(2)[1, 1, 1, 1]^T, U2 = (1)/(2)[1, 1, -1, -1]^T,
U3 = (1)/(2)[1, -1, 1, -1]^T, and U4 = (1)/(2)[1, -1, -1, 1]^T.
First, find c1, c2, c3, c4 for which
Y = c1U1 + c2U2 + c3U3 + c4U4.
The distribution of (c2^2 + c3^2 + c4^2)/(σ^2) is
F distribution
T distribution
Chi-square
Normal

Added by Victoria W.

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