Mean vector and covariance matrix

Question

I am given a home work for one subject, but my probability theory course is just started, so I dont have enough information. Could someone help me with that? Given: $$\begin{equation} p_\underline x(x)=\left\{ \begin{array}{@{}ll@{}} \frac1\pi, & \text{if}\ x^2_1+x^2_2 < 1 \\ 0, & \text{otherwise} \end{array}\right. \end{equation} $$ Find the mean and covariance matrix of the random vector of: $$ \underline y=\begin{bmatrix} 1 & -1 \\ 0 & 2 \\ \end{bmatrix}\underline x + \begin{bmatrix} 2 \\ 3 \\ \end{bmatrix} $$ Marginal distribution, mean and variance is already determined. Help me please with doing mean and covariance matrix. If this will be explained and possibly given a link to some resource it would be quite helpful for me. Thanx

Answer 1

Hint: not a complete solution, since you've already done most of the work, given what you've stated. Write

$$\underline{y} = \begin{bmatrix} y_1 \\ y_2 \end{bmatrix}, \underline{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\text{.}$$ Then we have $$\begin{bmatrix} y_1 \\ y_2 \end{bmatrix} = \begin{bmatrix} 1 & -1 \\ 0 & 2 \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} + \begin{bmatrix} 2 \\ 3 \end{bmatrix} = \begin{bmatrix} x_1 - x_2 + 2\\ 2x_2+3 \end{bmatrix}\text{.}$$ So, what this problem boils down to is finding the mean and covariance matrix of $$\begin{bmatrix} x_1 - x_2 + 2\\ 2x_2+3 \end{bmatrix}\text{.}$$ The expected value is simply $$\mathbb{E}\left[\begin{bmatrix} x_1 - x_2 + 2\\ 2x_2+3 \end{bmatrix} \right] = \begin{bmatrix} \mathbb{E}[x_1 - x_2 + 2]\\ \mathbb{E}[2x_2+3] \end{bmatrix}$$ and the covariance matrix is $$\text{Cov}\left(\begin{bmatrix} x_1 - x_2 + 2\\ 2x_2+3 \end{bmatrix}\right) = \begin{bmatrix} \text{Cov}\left(x_1 - x_2 + 2, x_1 - x_2 + 2\right) & \text{Cov}\left(x_1 - x_2 + 2, 2x_2+3\right) \\ \text{Cov}\left(2x_2+3, x_1 - x_2 + 2\right) & \text{Cov}\left(2x_2+3, 2x_2+3\right) \end{bmatrix}\text{.}$$ I will let you handle it from here.