Let \mathrm{{\boldsymbol X}} \sim \mathcal{N}( \mathbf{0}, \Sigma ) where, for simplicity, \mathrm{{\boldsymbol X}} \in \mathbb {R}^2 and hence \Sigma \in \mathbb {R}^{2 imes 2}.

Question

Let \mathrm{{\boldsymbol X}} \sim \mathcal{N}( \mathbf{0}, \Sigma ) where, for simplicity, \mathrm{{\boldsymbol X}} \in \mathbb {R}^2 and hence \Sigma \in \mathbb {R}^{2 	imes 2}.

We can express 	extsf{Var}(\mathrm{{\boldsymbol X}}^1) as \mathbf{u}^ T \Sigma \mathbf{u} for some unit vector \mathbf{u}.

What is \mathbf{u}?

\mathbf{u}^1 = 
 unanswered  , \mathbf{u}^2 = 
 unanswered 
Similarly, we can express 	extsf{Var}(\mathrm{{\boldsymbol X}}^2) as \mathrm{{\boldsymbol v}}^ T \Sigma \mathrm{{\boldsymbol v}} for some unit vector \mathrm{{\boldsymbol v}}.

What is \mathrm{{\boldsymbol v}}?

\mathrm{{\boldsymbol v}}^1 = 
 unanswered  , \mathrm{{\boldsymbol v}}^2 = 
 unanswered 
Finally, we can express 	extsf{Var}( \mathrm{{\boldsymbol X}}^1 + \mathrm{{\boldsymbol X}}^2 ) as \mathrm{{\boldsymbol w}}^ T \Sigma \mathrm{{\boldsymbol w}} for some vector \mathrm{{\boldsymbol w}}.

What is \mathrm{{\boldsymbol w}}?

\mathrm{{\boldsymbol w}}^1 = 
 unanswered  , \mathrm{{\boldsymbol w}}^2 = 
 unanswered

GPT 3.5 · Answer

To find the vectors u, v, and w, we need to use the properties of the covariance matrix.

Given that X ~ N(0, Σ), we know that the covariance matrix Σ is symmetric and positive semi-definite. Therefore, it can be represented as Σ = PΛP^T, where P is an orthogonal matrix composed of eigenvectors of Σ, and Λ is a diagonal matrix composed of the corresponding eigenvalues.

To find u and v, we need to normalize the eigenvectors of Σ. Let's denote the eigenvectors of Σ as p1 and p2, and the corresponding eigenvalues as λ1 and λ2.

Then, u = p1 / ||p1||, where ||p1|| is the Euclidean norm of p1. Similarly, v = p2 / ||p2||.

To find w, we need to consider the sum of two random variables. Let X1 and X2 be the two components of X.

The covariance matrix of the sum of two random variables is given by the sum of the individual covariance matrices, i.e., Cov(X1+X2) = Cov(X1) + Cov(X2). Therefore, the covariance matrix of the sum is Σ_sum = Σ_X1 + Σ_X2 = Σ.

Since w is a vector that represents the sum of X1 and X2, it can be written as w = (1, 1). So, w^T = (1, 1).

Therefore, the final answers are:

u^1 = p1^1 / ||p1||, u^2 = p1^2 / ||p1||
v^1 = p2^1 / ||p2||, v^2 = p2^2 / ||p2||
w^1 = 1, w^2 = 1

Let \mathrm{{\boldsymbol X}} \sim \mathcal{N}( \mathbf{0}, \Sigma ) where, for simplicity, \mathrm{{\boldsymbol X}} \in \mathbb {R}^2 and hence \Sigma \in \mathbb {R}^{2 \times 2}.