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}.
We can express \textsf{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 \textsf{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 \textsf{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
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
To find the values of u, v, and w, we need to first determine the covariance matrix Sigma.
Given that X ~ N(0, Sigma), the covariance matrix Sigma is defined as:
Sigma = E[(X - mu)(X - mu)^T],
where mu is the mean vector, which is 0 in this case.
In matrix form, Sigma = E[XX^T].
To find the entries of Sigma, we need to calculate E[XX^T].
Now, let's calculate the values of u, v, and w step-by-step.
Step 1: Calculate Var(X^1)
Var(X^1) = u^T * Sigma * u,
where u is a unit vector.
Step 2: Calculate Var(X^2)
Var(X^2) = v^T * Sigma * v,
where v is a unit vector.
Step 3: Calculate Var(X^1 + X^2)
Var(X^1 + X^2) = w^T * Sigma * w,
where w is a vector.
Let's calculate u, v, and w step-by-step.
Step 1: Calculate u
To find u, we need to calculate Var(X^1) first.
Since X^1 is the first element of X, it means that X^1 = X_1.
Thus, we need to calculate Var(X_1), which is the first entry of Sigma.
u^1 = 1 and u^2 = 0, such that u = [1, 0].
Step 2: Calculate v
To find v, we need to calculate Var(X^2) first.
Since X^2 is the second element of X, it means that X^2 = X_2.
Thus, we need to calculate Var(X_2), which is the second entry of Sigma.
v^1 = 0 and v^2 = 1, such that v = [0, 1].
Step 3: Calculate w
To find w, we need to calculate Var(X^1 + X^2) first.
(X^1 + X^2) = (X_1 + X_2).
Thus, we need to calculate Var(X_1 + X_2), which is the sum of the first and second entries of Sigma.
w^1 = 1 and w^2 = 1, such that w = [1, 1].
Therefore,
u = [1, 0],
v = [0, 1],
w = [1, 1].
To determine the vectors u, v, and w that satisfy the given expressions, we can start by considering the definition of the variance of a random variable.
The variance of a random variable X is defined as the expected value of the squared difference between X and its mean. In the case of a multivariate random variable X with mean μ and covariance matrix Σ, the variance can be expressed as Var(X) = (X - μ)^T Σ (X - μ).
Let's proceed to find the vectors u, v, and w.
1. Finding u:
To determine u, let's focus on Var(X^1). Since X^1 is the first component of X, we can express it as X^1 = [1, 0] X, where [1, 0] represents the unit vector along the first dimension.
Now, substituting X^1 = [1, 0] X in the variance expression, we have:
Var(X^1) = ([1, 0] X - μ)^T Σ ([1, 0] X - μ)
Comparing this expression with the given form Var(X^1) = u^T Σ u, we can conclude that u = [1, 0].
Therefore, u^1 = 1 and u^2 = 0.
2. Finding v:
Similarly, to determine v, let's consider Var(X^2). Since X^2 is the second component of X, we can express it as X^2 = [0, 1] X, where [0, 1] represents the unit vector along the second dimension.
Substituting X^2 = [0, 1] X in the variance expression, we have:
Var(X^2) = ([0, 1] X - μ)^T Σ ([0, 1] X - μ)
Comparing this expression with the given form Var(X^2) = v^T Σ v, we can conclude that v = [0, 1].
Therefore, v^1 = 0 and v^2 = 1.
3. Finding w:
To find w, let's consider Var(X^1 + X^2). We can express it as X^1 + X^2 = [1, 1] X, where [1, 1] represents the vector formed by adding the unit vectors along the first and second dimensions.
Substituting X^1 + X^2 = [1, 1] X in the variance expression, we have:
Var(X^1 + X^2) = ([1, 1] X - μ)^T Σ ([1, 1] X - μ)
Comparing this expression with the given form Var(X^1 + X^2) = w^T Σ w, we can conclude that w = [1, 1].
Therefore, w^1 = 1 and w^2 = 1.
To summarize:
u^1 = 1, u^2 = 0
v^1 = 0, v^2 = 1
w^1 = 1, w^2 = 1