Let's break down the problem step by step:

Given:

• A random variable $X$ with:
  • Mean $E[X] = 3$
  • Variance $\sigma_X^2 = 2$

You are asked to:

1. Determine the second moment of $X$ about the origin (i.e., $E[X^2]$).
2. Another random variable $Y = -6X + 22$:
   • Prove that $X$ and $Y$ are orthogonal.
   • Check if $X$ and $Y$ are uncorrelated.

Step 1: Second Moment of $X$ about the Origin

The second moment about the origin is simply $E[X^2]$. We know that the variance $\sigma_X^2$ is given by:

[ \sigma_X^2 = E[X^2] - (E[X])^2 ]

Plugging in the values:

$2 = E[X^2] - 3^2$

$2 = E[X^2] - 9$

Solving for $E[X^2]$:

$E[X^2] = 2 + 9 = 11$

So, the second moment of $X$ about the origin is $E[X^2] = 11$.

Step 2: Orthogonality of $X$ and $Y$

Two random variables are orthogonal if their inner product, which is $E[XY]$, is zero. To compute $E[XY]$:

$Y = -6X + 22$

So:

[ E[XY] = E[X(-6X + 22)] = -6E[X^2] + 22E[X] ]

Using the values we have:

$E[XY] = -6(11) + 22(3) = -66 + 66 = 0$

Since $E[XY] = 0$, $X$ and $Y$ are orthogonal.

Step 3: Uncorrelatedness of $X$ and $Y$

Two random variables are uncorrelated if their covariance is zero. The covariance is given by:

[ \text{Cov}(X, Y) = E[XY] - E[X]E[Y] ]

We already know that $E[XY] = 0$, so we just need to compute $E[Y]$. From the expression $Y = -6X + 22$, we find:

[ E[Y] = E[-6X + 22] = -6E[X] + 22 = -6(3) + 22 = -18 + 22 = 4 ]

Now, compute the covariance:

$\text{Cov}(X, Y) = 0 - (3)(4) = -12$

Since $\text{Cov}(X, Y) = -12$, $X$ and $Y$ are not uncorrelated.

Summary:

1. The second moment of $X$ about the origin is $E[X^2] = 11$.
2. $X$ and $Y$ are orthogonal.
3. $X$ and $Y$ are not uncorrelated (Covariance = -12).

Would you like more details on any step? Here are five related questions to expand on this:

1. What is the relationship between orthogonality and uncorrelatedness?
2. How does the variance formula relate to the second moment of a random variable?
3. What are the practical implications of two variables being orthogonal?
4. How would you compute the covariance of two general linear combinations of random variables?
5. Can orthogonality between two random variables imply independence?

Tip: Orthogonality often suggests independence in the case of Gaussian variables, but this is not true in general.