Q-function

In <a href="/facts/Statistics/DNPsKGYU">statistics</a>, the Q-function is the <a href="/facts/Cumulative_distribution_function/WaKU8tp4">tail distribution function</a> of the <a href="/facts/Normal_distribution/UapjjPyQ">standard normal distribution</a>. In other words, 
 
 
 
 Q
 (
 x
 )
 
 
 {\displaystyle Q(x)}
 
 is the probability that a normal (Gaussian) <a href="/facts/Random_variable/TwTBXnLT">random variable</a> 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 <a href="/facts/Normal_distribution/UapjjPyQ">standard normal</a> 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 <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution function</a>, are also used occasionally.
Because of its relation to the <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution function</a> of the normal distribution, the Q-function can also be expressed in terms of the <a href="/facts/Error_function/Ca2H7pGr">error function</a>, which is an important function in applied mathematics and physics.

Q-function open-in-new

Q-function