Var — variance

$\operatorname{Var}(X) = \mathbb{E}[X^2] - (\mathbb{E}[X])^2$: the average squared distance from the mean. Two rules are used repeatedly in the course:

• For independent quantities, variances add: $\operatorname{Var}(X + Y) = \operatorname{Var}(X) + \operatorname{Var}(Y)$.
• Scaling by a number $c$ scales the variance by $c^2$: $\operatorname{Var}(cX) = c^2 \operatorname{Var}(X)$.

If the entries of $\mathbf{q}$ and $\mathbf{k}$ are independent with mean 0 and variance 1, each product $q_i k_i$ has variance 1, so the first rule gives $\operatorname{Var}(\mathbf{q}^\top\mathbf{k}) = d_k$. The second rule then shows that dividing by $\sqrt{d_k}$ brings it back to 1.