This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

NormalDistribution

 NormalDistribution represents a normal (Gaussian) distribution with mean and standard deviation . NormalDistributionrepresents a normal distribution with zero mean and unit standard deviation.
• The probability density for value in a normal distribution is proportional to . »
Probability density function:
Cumulative distribution function:
Mean and variance:
Median:
Probability density function:
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Cumulative distribution function:
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Mean and variance:
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Median:
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 Scope   (6)
Generate a set of pseudorandom numbers that are normally distributed:
Compare its histogram to the PDF:
Distribution parameters estimation:
Estimate the distribution parameters from sample data:
Compare the density histogram of the sample with the PDF of the estimated distribution:
Skewness and kurtosis are constant:
Different moments with closed forms as functions of parameters:
Closed form for symbolic order:
Closed form for symbolic order:
Closed form for symbolic order:
Hazard function of a normal distribution is increasing:
Quantile function:
 Applications   (9)
Find the percentage of values that lie between and :
Between and :
Between and :
Package it up as a function:
Compute -values for a -test with alternative hypothesis :
Alternative hypothesis :
Alternative hypothesis :
A battery has a lifetime that is approximately normally distributed with a mean of 1000 hours and a standard deviation of 50 hours. Find the fraction with a lifetime between 800 and 1000 hours:
Out of 100 batteries, compute how many have a lifetime between 800 and 1000 hours:
Coffee beans are sold in 5 lb sacks that have true weight normally distributed with a mean of 5 lbs and a variance of 0.01 lb. Find the probability that a given sack weighs at least 4 lbs, 15 oz:
This can be directly computed from the SurvivalFunction:
A company manufactures nails with length normally distributed, mean 0.497 inches, and standard deviation 0.002 inches. Find the fraction that satisfies the specification of length equal to 0.5 inches plus/minus 0.004 inches:
Direct computation with CDF:
A company manufactures nails with length normally distributed and a mean of 0.5 inches. If 50% of the produced nails have lengths between 0.495 and 0.505, find the standard deviation:
Find the standard deviation:
A sample is selected from a distribution with mean 5 and standard deviation 1.5. Find the minimum size of the sample so that with probability 0.97 the sample mean is within 0.8 of the distribution mean:
The probability as a function of sample size:
Find the minimum sample size :
The weight of a person including luggage has normal distribution with mean 200 and standard deviation 40. A plane's load limit is 10000 lbs and it can take 42 passengers. With the maximum number of passengers on board, what is the probability of the plane being overloaded?
Normally distributed points in the plane:
Normally distributed points in 3D:
Parameter influence on the CDF for each :
Normal distribution is closed under translation and scaling:
In general, affine transformations of independent normals are normal:
The normal distribution is symmetric about its mean:
Relationships to other distributions:
Normal (SN) JohnsonDistribution is a normal distribution:
StudentTDistribution goes to a normal distribution as goes to :
Normal distribution is a transformation of LogNormalDistribution:
The inverse transformation of normal distribution yields LogNormalDistribution:
HalfNormalDistribution is a truncated normal distribution:
The normal and half-normal distributions:
HalfNormalDistribution is a transformation of normal distribution:
HalfNormalDistribution is a transformation of normal distribution:
Normal distribution is a special case of SkewNormalDistribution with shape parameter :
SkewNormalDistribution is a transformation of normal distribution:
Sum of squares of standard normally distributed variables follows ChiSquareDistribution:
Sum of squares of normally distributed variables has NoncentralChiSquareDistribution:
The norm of standard normally distributed variables follows ChiDistribution:
The norm of three standard normal variables has MaxwellDistribution, a case of ChiDistribution:
The norm of two standard normally distributed variables follows RayleighDistribution:
The norm of two normally distributed variables follows RiceDistribution:
NormalDistribution is the limiting case of HyperbolicDistribution of for and :
If , , , and are independent and normal, then has LaplaceDistribution:
If , , , and are independent and normal, then has LaplaceDistribution:
Quotient of two normally distributed variables has CauchyDistribution:
Square of a normally distributed variable is a special case of GammaDistribution, and also of ChiSquareDistribution:
LaplaceDistribution is a parameter mixture of a normal distribution with RayleighDistribution:
LevyDistribution is a transformation of a normal distribution:
With scale:
Normal distribution is a special case of type 3 PearsonDistribution:
Normal distribution is a StableDistribution:
Normal distribution is the marginal distribution of BinormalDistribution:
Normal distribution is the marginal distribution of MultinormalDistribution:
NormalDistribution is not defined when is not a real number:
NormalDistribution is not defined when is not a positive real number:
Substitution of invalid parameters into symbolic outputs gives results that are not meaningful:
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