GAUSSIAN/NORMAL
CUMULATIVE DISTRIBUTION TABLES
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- Used to find numerical values for Gaussian Cumulative Distribution Function because the Gaussian probability density function is not analytically integrable.
- The Cumulative Distribution Function of a Gaussian random variable is and can be illustrated as
The cumulative distribution function (CDF) of an Gaussian random variable is not available in a closed form, because the pdf equation is not integrable. Instead tables are used to find the value of this function at several values.
The tables come in three common forms. Each is the numerical integration of the unit normal (N[0,1]) Gaussian pdf but integrated over different regions.
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Phi(x)
is the proper CDF of the Gaussian:
Click here
for a table of values.
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Q(x)
Often the table of numbers is not for the integral above, but for the integral of Q(x). The relationship between Q(x) and is: The integral that describes Q(x) is:
Click here
for a table of Q(x) values.
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erf(x)
The other form of numerical integration results for the Gaussian CDF is the error function or "erf" function. erf(x) is the integral from 0 to x:
Click here
for a table of erf(x) values.
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Other Tables
These are not the only functions that are made available for use in integrating the Gaussian function. In MATLAB the erf() function is defined as To get the area under a standard Normal, the conversion Also sometimes the table of integrals for is given because the integral has fewer constants and the table can be used for multiple applications where an integral of that form is desired. Basically, always verify to yourself what the table is actually integrating before using it.
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A Useful Approximation
When you do not have a table handy, or your calculator or computer does not have a built-in erf(), you can still approximate these integrals. The approximation gives pretty good results. You should be able to convert this into a Phi(x) or an erf(x).
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When not N(0,1)
These tables are all for Gaussians with a mean of 0 and a standard deviation of 1. Providing tables for all possible means and standard deviations would not be practical. Fortunately it is possible to use any of these tables to find probabilities for Gaussian Random variables that have means different from 0 or standard deviations different from 1.
By using a transformation of variables making and thus substituting this into any of the above integrals gives a Gaussian of mean m and standard deviation s inside, and note that the limits of integration have changed too.
Also the functions Phi(x), Q(x) and erf(x) are for regions from to x, x to or 0 to x. When a probability for a region different that this is desired a linear combination of these must be used, although any single function can be used to find all probabilities.
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Definite Integrals of the form [ ]
When you need to integrate a definite integral of the above form, these Gaussian tables can be used. For a more complete discussion of this, click here.
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