In statistics, the Q function Q(x) usually refers to the normal distribution function Φ(x). For other meanings, see Other Uses.
The Normal Distribution Q Function Φ(x)
The CDF for the normal distribution gives you the probability that a normal random variable takes a value equal to or smaller than x. The Q function is the complement of this; In other words, it’s the probability a normal random variable takes a value greater than x.
As a formula:
Q(x) = 1 – CDF = P(X > x)
The plot starts with an area of 1, representing 100% probability. At the point on the far left of the bell curve, the right “tail” is actually the entire area of the curve.
Calculating the function by hand is relatively simple: find the CDF, and subtract from one. Some software programs find the Q function directly. For example, In MATLAB, the syntax is y = qfunc(x). If your software doesn’t, find the CDF and subtract from one.
There are other meanings for the Q function, including:
- The conditional expected log-likelihood, used in calculating the E-step in the EM algorithm (see Gupta and Chen 2011 for an example),
- The nome q, a special function used in the theory of elliptic functions.
- Q-analogs and Q-series, used in combinatorics and the study of functions.
- The q-products Qn, where n = 1,2,3.
- The partition function Q, used in statistical thermodynamics.
- The Marcum Q-function, which is used mainly in signal processing.
Gupta, M. & Chen, Y. (2011). Theory and Use of the EM Algorithm. Foundations and Trends in Signal Processing. Vol. 4, No. 3 (2010) 223–296. Retrieved December 6, 2017 from http://www.mayagupta.org/publications/EMbookGuptaChen2010.pdf
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