We assume that $\Sigma_Y$ satisfies $$\forall i \in \llbracket 1, n \rrbracket, \quad \sigma_{i,i} = \sigma^2 \quad \text{and} \quad \forall (i,j) \in \llbracket 1, n \rrbracket^2, \quad i \neq j \Longrightarrow \sigma_{i,j} = \sigma^2 \gamma$$ where $\sigma$ and $\gamma$ are two strictly positive real numbers. We denote by $J \in \mathcal{M}_n(\mathbb{R})$ the matrix whose coefficients are all equal to 1, and $U_0$ is a unit vector such that the variance of $Z = U_0^\top Y$ is maximal. Calculate the percentage of total variance represented by $Z$, that is, the ratio $\dfrac{\mathbb{V}(Z)}{\mathbb{V}_T(Y)}$.
We assume that $\Sigma_Y$ satisfies
$$\forall i \in \llbracket 1, n \rrbracket, \quad \sigma_{i,i} = \sigma^2 \quad \text{and} \quad \forall (i,j) \in \llbracket 1, n \rrbracket^2, \quad i \neq j \Longrightarrow \sigma_{i,j} = \sigma^2 \gamma$$
where $\sigma$ and $\gamma$ are two strictly positive real numbers. We denote by $J \in \mathcal{M}_n(\mathbb{R})$ the matrix whose coefficients are all equal to 1, and $U_0$ is a unit vector such that the variance of $Z = U_0^\top Y$ is maximal.
Calculate the percentage of total variance represented by $Z$, that is, the ratio $\dfrac{\mathbb{V}(Z)}{\mathbb{V}_T(Y)}$.