Skip to content Skip to navigation

OpenStax-CNX

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

Navigation

Recently Viewed

This feature requires Javascript to be enabled.
 

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 .

Aside:

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
(1)

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

Approximations

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

Content actions

Download module as:

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

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

What is in a lens?

Lens makers point to 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 member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks