Average of Block Vectors

Question

Let $x$ be a block vector with two vector elements, $x=(a,b)$ where $a$ and $b$ are vectors of size $n$ and $m$, respectively.

Express $avg(x)$ in terms of $avg(a)$, $avg(b)$, $m$, and $n$.

Answer 1

By $\text{avg}(x)$, I presume you mean the mean of the elements of $x$. If so, then \begin{equation*} \text{avg}(x) = \frac{1}{n+m}\sum_{i=1}^{n+m}x_i = \frac{1}{n+m}\left(\sum_{i=1}^n a_i + \sum_{j=1}^m b_j\right) = \frac{1}{n+m}\left( n\text{avg}(a) + m\text{avg}(b) \right). \end{equation*}