In statistics, the Q-function is the tail distribution function of the standard normal distribution. In other words, Q ( x ) {\displaystyle Q(x)} is the probability that a normal (Gaussian) random variable will obtain a value larger than x {\displaystyle x} standard deviations. Equivalently, Q ( x ) {\displaystyle Q(x)} is the probability that a standard normal random variable takes a value larger than x {\displaystyle x} . If Y {\displaystyle Y} is a Gaussian random variable with mean μ {\displaystyle \mu } and variance σ 2 {\displaystyle \sigma ^{2}} , then X = Y − μ σ {\displaystyle X={\frac {Y-\mu }{\sigma }}} is standard normal and
P ( Y > y ) = P ( X > x ) = Q ( x ) {\displaystyle P(Y>y)=P(X>x)=Q(x)}
where x = y − μ σ {\displaystyle x={\frac {y-\mu }{\sigma }}} . Other definitions of the Q-function, all of which are simple transformations of the normal cumulative distribution function, are also used occasionally. Because of its relation to the cumulative distribution function of the normal distribution, the Q-function can also be expressed in terms of the error function, which is an important function in applied mathematics and physics.
Definition and basic properties Formally, the Q-function is defined as
Q ( x ) = 1 2 π ∫ x ∞ exp ( − u 2 2 ) d u . {\displaystyle Q(x)={\frac {1}{\sqrt {2\pi }}}\int _{x}^{\infty }\exp \left(-{\frac {u^{2}}{2}}\right)\,du.}
Thus,
Q ( x ) = 1 − Q ( − x ) = 1 − Φ ( x ) , {\displaystyle Q(x)=1-Q(-x)=1-\Phi (x)\,\!,}
where Φ ( x ) {\displaystyle \Phi (x)} is the cumulative distribution function of the standard normal Gaussian distribution. The Q-function can be expressed in terms of the error function, or the complementary error function, as
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