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The Q-function

Module by: Clayton Scott, Robert Nowak. E-mail the authors

The QQ-function is a convenient way to express right-tail probabilities for normal (Gaussian) random variables. For xR x , Qx Q x is defined as the probability that a standard normal random variable (zero mean, unit variance) exceeds xx: Qx=12πxet22d t Q x 1 2 t x t 2 2 QQ is a mapping from R to 01 0 1 . One may also define Q=1 Q 1 and Q=0 Q 0 .


Some authors define the QQ-function in different ways. One alternative is to define Qx=FxF0 Q x F x F 0 . This definition is discussed at MathWorld.
Figure 1: Qx Q x is represented by the shaded region.
Figure 1 (.png)
Since Qx Q x is monotonically decreasing, it has a well-defined inverse Q-1 : 01 R Q : 0 1 .
Figure 2: A plot of Qx Q x
Figure 2 (.png)
If Fx F x denotes the cumulative distribution function of a standard normal, then clearly Qx=1Fx Q x 1 F x . For this reason QQ is also called the complementary cumulative distribution function. The QQ-function is useful because the tail probability cannot be evaluated symbolically, and so Qx 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 QQ-function is also useful for expressing right-tail probabilities of nonstandard normal variates. If X𝒩μσ2 X μ σ 2 then Xμσ𝒩01 X μ σ 0 1 To express PrX>γ X γ in terms of QQ, where γR γ , define η=γμσ η γ μ σ . Then

PrX>γ=PrX>ησ+μ=PrXμσ>η=Qη=Qγμσ X γ X η σ μ X μ σ η Q η Q γ μ σ

Relation to Erf and Erfinv

The erferf function is defined as erfx=2π0xet2d t erf x 2 t 0 x t 2 Both erferf and its inverse, erfinverfinv, are built into many common mathematical software packages such as Mathematica and Matlab. Therefore, they can be used to numerically evaluate QQ and Q-1 Q . By a change of variables, we have Qx=12(1erfx2) Q x 1 2 1 erf x 2 and Q-1y=2erfinv12y Q y 2 erfinv 1 2 y


One approximation that is sometimes useful for xx away from zero is Qx1x2πe-12x2 Q x 1 x 2 -1 2 x 2

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