Calculating Covariance with a Random Vector

[ElijahRockers]

Homework Statement

Let $X$ be a random variable such that $\mu_X = 0$ and $K_{XX} = I$.
Find $Cov(a^T X, b^T X)$ for $a = (1, 1, 0, 0)$ and $b = (0, 1, 1, 0)$.

The Attempt at a Solution

I guess I am assuming that $X$ is a 4 element random vector. I can't know values of the random variables, but I know their mean, and I think from $K_{XX} = I$ that
$E[X_i X_j] = 0, i≠ j$
$E[X_i X_j] = 1, i= j$

So..

$a^T X = X_1 + X_2 = A$
$b^T X = X_2 + X_3 = B$
$Cov(A,B) = E[AB]-E[A]E[ B]$

$E[A]$ and $E[ B]$ are 0, so

$Cov(A,B) = E[AB] = E[X_1 X_2 + X_1 X_3 + X_2 X_2 + X_2 X_3]$

From $K_{XX}$, $E[AB] = E[X_2 X_2] = 1 = Cov(A,B)$

Not sure if this is correct or not.

 

[Ray Vickson]

Homework Statement

Let $X$ be a random variable such that $\mu_X = 0$ and $K_{XX} = I$.
Find $Cov(a^T X, b^T X)$ for $a = (1, 1, 0, 0)$ and $b = (0, 1, 1, 0)$.

The Attempt at a Solution

I guess I am assuming that $X$ is a 4 element random vector. I can't know values of the random variables, but I know their mean, and I think from $K_{XX} = I$ that
$E[X_i X_j] = 0, i≠ j$
$E[X_i X_j] = 1, i= j$

So..

$a^T X = X_1 + X_2 = A$
$b^T X = X_2 + X_3 = B$
$Cov(A,B) = E[AB]-E[A]E[ B]$

$E[A]$ and $E[ B]$ are 0, so

$Cov(A,B) = E[AB] = E[X_1 X_2 + X_1 X_3 + X_2 X_2 + X_2 X_3]$

From $K_{XX}$, $E[AB] = E[X_2 X_2] = 1 = Cov(A,B)$

Not sure if this is correct or not.

It is correct if your interpretation of $K_{XX}$ is correct (which I cannot speak to because the notation is unfamiliar to me).

 

[ElijahRockers]

It is correct if your interpretation of $K_{XX}$ is correct (which I cannot speak to because the notation is unfamiliar to me).

$K_{XX}$ is the covariance matrix of $X$, where $K_{XX_{i,j}} = E[X_i X_j] - E[X_i]E[X_j]$ is each element in the matrix... I believe.