| 释义 |
Gaussian DistributionThe Gaussian probability distribution with Mean  and Standard Deviation  is a GaussianFunction of the form
 | (1) |
 gives the probability that a variate with a Gaussian distribution takes on a value in the range  . This distributionis also called the Normal Distribution or, because of its curved flaring shape, the Bell Curve. The distribution  is properly normalized for  since
 | (2) |
 , is then
 | (3) |
Gaussian distributions have many convenient properties, so random variates with unknown distributions are often assumedto be Gaussian, especially in physics and astronomy. Although this can be a dangerous assumption, it is often a goodapproximation due to a surprising result known as the Central Limit Theorem. This theorem states that theMean of any set of variates with any distribution having a finite Mean and Variance tends to theGaussian distribution. Many common attributes such as test scores, height, etc., follow roughly Gaussian distributions,with few members at the high and low ends and many in the middle.
Making the transformation
 | (4) |
 gives a variate with unit Variance and 0 Mean
 | (5) |
 is known as a z-Score.
The Normal Distribution Function gives the probability that a standard normal variate assumes a value in the interval ,
 | (6) |
 nor Erf can be expressed in termsof finite additions, subtractions, multiplications, and root extractions, and so both must be either computed numerically orotherwise approximated. The value of  for which falls within the interval  with a given probability  iscalled the Confidence Interval.
The Gaussian distribution is also a special case of the Chi-Squared Distribution, since substituting
 | (7) |
 | (8) |
 since runs from 0 to  instead of from  to ), gives
which is a Chi-Squared Distribution in  with  (i.e., a Gamma Distribution with  and  ).
Cramer showed in 1936 that if  and  are Independent variates and  has aGaussian distribution, then both and must be Gaussian (Cramer's Theorem).
The ratio  of independent Gaussian-distributed variates with zero Mean is distributed with a CauchyDistribution. This can be seen as follows. Let and both have Mean 0 and standard deviations of  and , respectively, then the joint probability density function is the Gaussian Bivariate Distribution with ,
 | (10) |
 is
But
 | (12) |
which is a Cauchy Distribution with Mean  and full width
 | (14) |
The Characteristic Function for the Gaussian distribution is
 | (15) |
Completing the Square in the exponent,
 | (17) |
The integral then becomes
so
and
These can also be computed using
yielding, as before,
The moments can also be computed directly by computing the Moments about the origin  ,
 | (31) |
giving
 | (35) |
where  are Gaussian Integrals.
Now find the Moments about the Mean,
so the Variance, Standard Deviation, Skewness, and Kurtosis are given by
The Variance of the Sample Variance  for a sample taken from a population with a Gaussian distribution is
If , this expression simplifies to
 | (50) |
 | (51) |
The Cumulant-Generating Function for a Gaussian distribution is
 | (52) |
For Gaussian variates,  for  , so the variance of k-Statistic  is
Also,
where
If is a Gaussian distribution, then
 | (62) |
 with a Gaussian distribution can be generated from variates  having a Uniform Distribution in(0,1) via
 | (63) |
The Gaussian distribution is an approximation to the Binomial Distribution in the limit of large numbers,
 | (64) |
 is the number of steps in the Positive direction,  is the number of trials ( ), and  and  are the probabilities of a step in the Positive direction and Negative direction ( ).
The differential equation having a Gaussian distribution as its solution is
 | (65) |
 | (66) |
 | (67) |
 | (68) |
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 533-534, 1987.Kraitchik, M. ``The Error Curve.'' §6.4 in Mathematical Recreations. New York: W. W. Norton, pp. 121-123, 1942. Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, p. 109-111, 1992. |
