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[[Category:Continuous distributions]] |
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[[Category:Continuous distributions]] |
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[[Category:Special functions]] |
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[[Category:Special functions]] |
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[[Category:Articles containing proofs]] |
In statistics, the Q-function is the tail probability of the standard normal distribution.[1][2] In other words, is the probability that a standard normal random variable will obtain a value larger than . Other definitions of the Q-function, all of which are simple transformations of the normal cumulative distribution function, are also used occasionally.[3]
Definition and basic properties
Formally, the Q-function is defined as
Thus,
where is the cumulative distribution function of the normal Gaussian distribution.
The Q-function can be expressed in terms of the error function as[2]
Bounds
The Q-function is not an elementary function. However, the bounds
become increasingly tight for large x, and are often useful.
Using the substitution and defining the upper bound is derived as follows:
Similarly, using and the quotient rule,
Solving for provides the lower bound.
Values
Some values of the Q-function are given below for reference.
Q(0.0) = 0.500000000
Q(0.1) = 0.460172163
Q(0.2) = 0.420740291
Q(0.3) = 0.382088578
Q(0.4) = 0.344578258
Q(0.5) = 0.308537539
Q(0.6) = 0.274253118
Q(0.7) = 0.241963652
Q(0.8) = 0.211855399
Q(0.9) = 0.184060125
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Q(1.0) = 0.158655254
Q(1.1) = 0.135666061
Q(1.2) = 0.115069670
Q(1.3) = 0.096800485
Q(1.4) = 0.080756659
Q(1.5) = 0.066807201
Q(1.6) = 0.054799292
Q(1.7) = 0.044565463
Q(1.8) = 0.035930319
Q(1.9) = 0.028716560
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Q(2.0) = 0.022750132
Q(2.1) = 0.017864421
Q(2.2) = 0.013903448
Q(2.3) = 0.010724110
Q(2.4) = 0.008197536
Q(2.5) = 0.006209665
Q(2.6) = 0.004661188
Q(2.7) = 0.003466974
Q(2.8) = 0.002555130
Q(2.9) = 0.001865813
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Q(3.0) = 0.001349898
Q(3.1) = 0.000967603
Q(3.2) = 0.000687138
Q(3.3) = 0.000483424
Q(3.4) = 0.000336929
Q(3.5) = 0.000232629
Q(3.6) = 0.000159109
Q(3.7) = 0.000107800
Q(3.8) = 0.000072348
Q(3.9) = 0.000048096
Q(4.0) = 0.000031671
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See also
References