Skip to content Skip to navigation

Connexions

You are here: Home » Content » The Q-function

Navigation

Content Actions
  • Download module PDF
  • Add to ...
    Add the module to:
    • My Favorites
    • A lens
    • An external social bookmarking service
    • My Favorites (What is 'My Favorites'?)
      'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.
    • A lens (What is a lens?)

      Definition of a lens

      Lenses

      A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

      What is in a lens?

      Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

      Who can create a lens?

      Any individual Connexions member, a community, or a respected organization.

    • External bookmarks
  • E-mail the authors

Recently Viewed

This feature requires Javascript to be enabled.

The Q-function

Module by: Clayton Scott, Robert Nowak

The QQ-function is a convenient way to express right-tail probabilities for normal (Gaussian) random variables. For x∈ℝ x , Qx Q x is defined as the probability that a standard normal random variable (zero mean, unit variance) exceeds xx: Qx=12π∫x∞ⅇ-t22dt 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 .

Aside:

Some authors define the QQ-function in different ways. One alternative is to define Qx=Fx-F0 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=1-Fx 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 X∼μσ2 X μ σ 2 then X-μσ∼01 X μ σ 0 1 To express PrX>γ X γ in terms of QQ, where γ∈ℝ γ , define η=γ-μσ η γ μ σ . Then

PrX>γ=PrX>ησ+μ=PrX-μσ>η=Qη=Qγ-μσ X γ X η σ μ X μ σ η Q η Q γ μ σ (1)

Relation to Erf and Erfinv

The erferf function is defined as erfx=2π∫0xⅇ-t2dt 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=121-erfx2 Q x 1 2 1 erf x 2 and Q-1y=2erfinv1-2y Q y 2 erfinv 1 2 y

Approximations

One approximation that is sometimes useful for xx away from zero is Qx≈1x2πⅇ-12x2 Q x 1 x 2 -1 2 x 2

Comments, questions, feedback, criticisms?

Send feedback