1.17 The q-function

The $Q$ -function is a convenient way to express right-tail probabilities for normal (Gaussian) randomvariables. For $x\in \mathbb{R}$ , $Q(x)$ is defined as the probability that a standard normal random variable (zero mean, unit variance) exceeds $x$ : $$Q(x)=\frac{1}{\sqrt{2\pi }}\int_x \,d t$$∞ t 2 2 $Q$ is a mapping from $\mathbb{R}$ to $\{0, 1\}$ . One may also define $Q(())$∞ 1 and $Q()$∞ 0 .

Arbitrary mean and variance

The $Q$ -function is also useful for expressing right-tail probabilities of nonstandard normalvariates. If $$(X, (, ^2))$$ then $$(\frac{X-}{}, (0, 1))$$ To express $(X> )$ in terms of $Q$ , where $\in \mathbb{R}$ , define $=\frac{-}{}$ . Then

Relation to erf and erfinv

The $\mathrm{erf}$ function is defined as $$\mathrm{erf}(x)=\frac{2}{\sqrt{\pi }}\int_0^x e^{-t^2}\,d t$$ Both $\mathrm{erf}$ and its inverse, $\mathrm{erfinv}$ , are built into many common mathematical software packages such asMathematica and Matlab. Therefore, they can be used to numerically evaluate $Q$ and $Q^{(-1)}$ . By a change of variables, we have $$Q(x)=\frac{1}{2}(1-\mathrm{erf}(\frac{x}{\sqrt{2}}))$$ and $$Q(y)^{(-1)}=\sqrt{2}\mathrm{erfinv}(1-2y)$$

Approximations

One approximation that is sometimes useful for $x$ away from zero is $$Q(x)\approx \frac{1}{x\sqrt{2\pi }}e^{\frac{-1}{2}x^2}$$