Upper bound for Variance of linear combination of random variables: $\operatorname{Var}\left(x^Ta\right) \leq \frac{\|a\|^2}{4}. $

Question

I found this while reading a paper where they used it as a casual fact.

Say, you have a vector $x = (x_1, x_2, \dots, x_n)$ where $x_i \in [0,1]$ are independent random variables. Consider linear combination $\sum_{i=1}^n x_ia_i$ and let $a = (a_1,a_2,\dots,a_n)$. Then we have

$$ \operatorname{Var}\left(x^Ta\right) \leq \frac{\|a\|^2}{4}. $$

I tried by using the following:

$$\operatorname{Var}\left(x^Ta\right) = \sum_{i=1}^n a_i^2 \operatorname{Var}(x_i),$$ but couldn't progress much from this point onward.

Answer 1

Let $X$ be a random variable such that $X\in[0,1]$ then $X^2\leq X$ so $$Var(X)=EX^2-(EX)^2\leq EX-(EX)^2=\frac14-\left(EX-\frac12\right)^2\leq \frac14.$$