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Q Function

In probability theory and statistics, the Q function is the tail probability of the standard normal distribution, also known as the complementary cumulative distribution function (CCDF) of the standard normal distribution. The Q function is defined as:

Q(x)=12πxet22dtQ(x) = \frac{1}{\sqrt{2\pi}} \int_{x}^{\infty} e^{-t^2\over2} dt

where x is a real number.

The Q function calculates the probability that a random variable from a normal distribution with mean 0 and standard deviation 1 exceeds a certain value, xx. This probability can be written as:

P(Z>x)=Q(x)P(Z>x) = Q(x)

where Z is a standard normal variable.

The Q function can be calculated using numerical integration or special functions, such as the complementary error function or the Marcum Q-function.

The Q function is helpful in various applications, such as digital communications and signal processing, where it calculates error probabilities and signal-to-noise ratios. It is also used in statistics to calculate confidence intervals and hypothesis tests for normal distributions.

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