I would like to write an equation that represents calculating the variance between a number of real-valued functions.
Let's say I have $n$ functions $Q_i: \mathbb{R} \rightarrow \mathbb{R}$, $i \in \{1,\dots,n\}$. I want to say that I should calculate the variance between the values of all $Q$ functions for a given input. Right now I have something like this:
$u(s) = var(\cup_{i=1}^n Q_i(s))$.
However, I am not sure if the $\cup$ operator is appropriate for this purpose, which symbol should I use?
Interesting.. I can think of a couple of options off the bat
The brute force method (equivalent to yours), variance of a set: $$u(s) = var\left( \{Q_k(s)\}_{k=1}^{n} \right) = var(\{ Q_1(s), Q_2(s), ..., Q_n(s) \})$$
Vector notation: Let $Q(s)$ be the vector of length $n$ defined as $$Q(s)=(Q_k(s))_{1 \leq k \leq n} = \begin{bmatrix} Q_1(s) \\ Q_2(s) \\ \vdots \\ Q_n(s) \end{bmatrix}$$ Then you can define variance of a vector and denote it as $var(Q)$