The Q-function 1.2 2003/08/01 10:55:39 GMT-5 2003/08/06 14:42:44.059 GMT-5 Rob "The Kid" Nowak nowak@rice.edu Clayton Scott cscott@rice.edu Rob "The Kid" Nowak nowak@rice.edu Clayton Scott cscott@rice.edu Elizabeth Gregory lizzardg@rice.edu Jeffrey Silverman jsilv@rice.edu complementary cumulative distribution function The Q-function is a convenient way to express right-tail probabilities for normal (Gaussian) random variables. For x , Q x is defined as the probability that a standard normal random variable (zero mean, unit variance) exceeds x: Q x 1 2 t x t 2 2 Q is a mapping from to 0 1 . One may also define Q 1 and Q 0 . Some authors define the Q-function in different ways. One alternative is to define Q x F x F 0 . This definition is discussed at MathWorld.

Q x is represented by the shaded region.

Since Q x is monotonically decreasing, it has a well-defined inverse Q : 0 1 → .

A plot of Q x

If F x denotes the cumulative distribution function of a standard normal, then clearly Q x 1 F x . For this reason Q is also called the complementary cumulative distribution function. The Q-function is useful because the tail probability cannot be evaluated symbolically, and so Q x offers a concise notation for this integral. It is similar to the gamma and beta functions in this respect.

Arbitrary Mean and Variance The Q-function is also useful for expressing right-tail probabilities of nonstandard normal variates. If X μ σ 2 then X μ σ 0 1 To express X γ in terms of Q, where γ , define η γ μ σ . Then X γ X η σ μ X μ σ η Q η Q γ μ σ

Relation to Erf and Erfinv The erf function is defined as erf x 2 t 0 x t 2 Both erf and its inverse, erfinv, are built into many common mathematical software packages such as Mathematica and Matlab. Therefore, they can be used to numerically evaluate Q and Q . By a change of variables, we have Q x 1 2 1 erf x 2 and Q y 2 erfinv 1 2 y

Approximations One approximation that is sometimes useful for x away from zero is Q x 1 x 2 -1 2 x 2