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NormalDistribution
✖
NormalDistribution

✖

represents a normal (Gaussian) distribution with mean μ and standard deviation σ.

✖

represents a normal distribution with zero mean and unit standard deviation.

Details

Background & Context

  • NormalDistribution[μ,σ] represents the so-called "normal" statistical distribution that is defined over the real numbers. The distribution is parametrized by a real number μ and a positive real number σ, where μ is the mean of the distribution, σ is known as the standard deviation, and σ2 is known as the variance. The probability density function (PDF) of a normal distribution is unimodal, with the peak occurring at the mean , and the parameter σ determines both the height of the PDF and the "thickness" of its tails. The PDF of a normal distribution is symmetric about its maximum, and the tails of its PDF are "thin" in the sense that the PDF decreases exponentially for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) The zero-argument form NormalDistribution[] is equivalent to NormalDistribution[0,1] and is sometimes called the standard normal distribution.
  • Due to the presence of the Gaussian function in its PDF, a normal distribution is sometimes referred to as a Gaussian distribution. Informally, a normal distribution may also be referred to as a "bell curve" as a result of the bell-like shape of its PDF. However, it should be noted that other distributions such as CauchyDistribution, StudentTDistribution, and LogisticDistribution also display qualitatively similar "bell" shapes.
  • Random variables that are normally distributed are sometimes called normal variates, and the standard normal distribution may also be referred to as the unit normal distribution.
  • Normal distributions are among the most widely occurring probability distributions and thus have many applications. For example, normally distributed values are of fundamental importance in applications of the Monte Carlo method. In addition, the normal distribution is also fundamental in defining the so-called Wiener process, a continuous-time stochastic process consisting of independent increments , each of which is independent and identically normally distributed with and for . Moreover, a number of probabilistic and statistical values including percentile ranks and - and -scores are derived from normal distributions. Furthermore, because of the central limit theorem, the mean of a sufficiently large number of independent random variables will be approximately normally distributed provided certain hypotheses are satisfied, regardless of the original distributions describing the variables. The normal distribution also arises naturally when modeling a number of physical phenomena, such as the velocity of ideal gas molecules, the positions of particles experiencing diffusion, and the long-timescale behavior of thermal light. In addition, a large number of biological phenomena, including sizes of living tissue and quantities such as fasting blood glucose level and blood pressure, yield variables whose logarithms tend to be normally distributed.
  • RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a normal distribution. Distributed[x,NormalDistribution[μ,σ]], written more concisely as xNormalDistribution[μ,σ], can be used to assert that a random variable x is distributed according to a normal distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
  • The probability density and cumulative distribution functions may be given using PDF[NormalDistribution[μ,σ],x] and CDF[NormalDistribution[μ,σ],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
  • DistributionFitTest can be used to test if a given dataset is consistent with a normal distribution, EstimatedDistribution to estimate a normal parametric distribution from given data, and FindDistributionParameters to fit data to a normal distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic normal distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic normal distribution.
  • TransformedDistribution can be used to represent a transformed normal distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a normal distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving normal distributions.
  • NormalDistribution is closely related to a number of other distributions. A number of distributions, including LogNormalDistribution, HalfNormalDistribution, NoncentralChiSquareDistribution, and LevyDistribution, can be viewed as transformed versions of NormalDistribution, while NormalDistribution can also be considered a limiting case of a number of distributions, including HyperbolicDistribution, StudentTDistribution, PoissonDistribution, and BinomialDistribution. In addition, NormalDistribution is a special case of ExponentialPowerDistribution (PDF[NormalDistribution[μ,σ],x] is the same as PDF[ExponentialPowerDistribution[2,μ,σ],x]), SkewNormalDistribution (PDF[NormalDistribution[μ,σ],x] is the same as PDF[SkewNormalDistribution[μ,σ,0],x]), and PearsonDistribution (PDF[NormalDistribution[μ,σ],x] is the same as PDF[PearsonDistribution[3,σ-2,-μ σ-2,0,0,1],x] when σ>0) and is the marginal distribution of BinormalDistribution and MultinormalDistribution. NormalDistribution is also closely related to StableDistribution, RiceDistribution, RayleighDistribution, MaxwellDistribution, LevyDistribution, LaplaceDistribution, JohnsonDistribution, ChiDistribution, and ChiSquareDistribution.

Examples

open allclose all

Basic Examples  (4)Summary of the most common use cases

Probability density function:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-7els8w
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-miefl7
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfuuto891sq-0i7ild
Out[3]=3

Cumulative distribution function:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-rcrn4t
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-675i6j
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfuuto891sq-7ojn4s
Out[3]=3

Mean and variance:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-sxr
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-g5k
Out[2]=2

Median:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-skc5b
Out[1]=1

Scope  (7)Survey of the scope of standard use cases

Generate a sample of pseudorandom numbers from a normal distribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-qhtk5j

Compare its histogram to the PDF:

In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-03mwaz
Out[2]=2

Distribution parameters estimation:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-45b7g2

Estimate the distribution parameters from sample data:

In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-epi747
Out[2]=2

Compare the density histogram of the sample with the PDF of the estimated distribution:

In[3]:=3
✖
https://wolfram.com/xid/0gfuuto891sq-f8ui5o
Out[3]=3

Skewness and kurtosis are constant:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-ezm
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-fme
Out[2]=2

Different moments with closed forms as functions of parameters:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-js043h

Moment:

In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-rx074o
Out[2]=2

Closed form for symbolic order:

In[3]:=3
✖
https://wolfram.com/xid/0gfuuto891sq-4bz2go
Out[3]=3

CentralMoment:

In[4]:=4
✖
https://wolfram.com/xid/0gfuuto891sq-pknsqa
Out[4]=4

Closed form for symbolic order:

In[5]:=5
✖
https://wolfram.com/xid/0gfuuto891sq-ut0n6s
Out[5]=5

FactorialMoment:

In[6]:=6
✖
https://wolfram.com/xid/0gfuuto891sq-zg9ct4
Out[6]=6

Cumulant:

In[7]:=7
✖
https://wolfram.com/xid/0gfuuto891sq-9gzmth
Out[7]=7

Closed form for symbolic order:

In[8]:=8
✖
https://wolfram.com/xid/0gfuuto891sq-7bvskj
Out[8]=8

Hazard function of a normal distribution is increasing:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-fafby6
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-h6jaaf
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfuuto891sq-d6k2f
Out[3]=3

Quantile function:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-yp5dg6
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-w11np
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfuuto891sq-vq6wt5
Out[3]=3

Consistent use of Quantity in parameters yields QuantityDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-bf9qsq
Out[1]=1

Find height quartiles:

In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-0vdsl
Out[2]=2

Applications  (11)Sample problems that can be solved with this function

Find the percentage of values that lie between and :

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-gl1aql
Out[1]=1

Between and :

In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-w905w5
Out[2]=2

Between and :

In[3]:=3
✖
https://wolfram.com/xid/0gfuuto891sq-p511ms
Out[3]=3

Package it up as a function:

In[4]:=4
✖
https://wolfram.com/xid/0gfuuto891sq-mzk8rj
In[5]:=5
✖
https://wolfram.com/xid/0gfuuto891sq-7t1ez0
Out[5]=5

Compute ‐values for a ‐test under the null hypothesis and the alternative :

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-fv1gym
Out[1]=1

Alternative hypothesis :

In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-pwgdil
Out[2]=2

Alternative hypothesis :

In[3]:=3
✖
https://wolfram.com/xid/0gfuuto891sq-o1rm7n
Out[3]=3

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:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-mqx4j
Out[1]=1

Out of 100 batteries, compute how many have a lifetime between 800 and 1000 hours:

In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-pxpvdx
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfuuto891sq-3ig8k
Out[3]=3

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:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-ceh9dk
Out[1]=1

This can be directly computed from the SurvivalFunction:

In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-iry0zc
Out[2]=2

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:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-mx5o44
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-hgy6it
In[3]:=3
✖
https://wolfram.com/xid/0gfuuto891sq-ctgl1d
Out[3]=3

Direct computation with CDF:

In[4]:=4
✖
https://wolfram.com/xid/0gfuuto891sq-djp2vg
Out[4]=4

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:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-eykifl
Out[1]=1

Find the standard deviation:

In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-gmmiaw
Out[2]=2

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:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-dxiws3
Out[1]=1

The probability as a function of sample size:

In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-iozf0
Out[2]=2

Find the minimum sample size :

In[3]:=3
✖
https://wolfram.com/xid/0gfuuto891sq-bhijjr
Out[3]=3

The weight of a person, including luggage, has normal distribution with mean 225 lbs and standard deviation 50 lbs. A plane's load limit is 10000 lbs and it can take 44 passengers. With the maximum number of passengers on board, find the probability of the plane being overloaded:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-d3nkva
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-dlqw1f
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfuuto891sq-cekcxe
Out[3]=3

Normally distributed points in the plane:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-m2yr2n
Out[1]=1

Normally distributed points in 3D:

In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-hxvgk0
Out[2]=2

Normal distribution was traditionally used to analyze the fractional stock price changes from the previous closing price. Find the estimated distribution for the daily fractional price changes of the S&P 500 index from January 1, 2000, to January 1, 2009:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-dmlawq
Out[1]=1

The range of fractional prices falls within the range of the normal distribution:

In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-qka0gn
Out[2]=2

Fit normal distribution:

In[3]:=3
✖
https://wolfram.com/xid/0gfuuto891sq-vmzp0a
Out[3]=3

Compare the histogram of the data with the PDF of the estimated distribution:

In[4]:=4
✖
https://wolfram.com/xid/0gfuuto891sq-dmgmv6
Out[4]=4

Find the probability of the fractional price change being greater than 0.5%:

In[5]:=5
✖
https://wolfram.com/xid/0gfuuto891sq-nwvp7k
Out[5]=5

Find the mean fractional price change:

In[6]:=6
✖
https://wolfram.com/xid/0gfuuto891sq-kp7n55
Out[6]=6

Simulate fractional price changes for 30 days:

In[7]:=7
✖
https://wolfram.com/xid/0gfuuto891sq-exrtb
Out[7]=7

Show that using LogisticDistribution provides better fit than using normal distribution:

In[8]:=8
✖
https://wolfram.com/xid/0gfuuto891sq-iz4hry
Out[8]=8
In[9]:=9
✖
https://wolfram.com/xid/0gfuuto891sq-sbgb9v
Out[9]=9

Generate Gaussian white noise:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-gfeazz
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-d6oki
In[3]:=3
✖
https://wolfram.com/xid/0gfuuto891sq-d1z61k
Out[3]=3

Properties & Relations  (36)Properties of the function, and connections to other functions

Normal distribution is closed under translation and scaling:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-lvbv97
Out[1]=1

In general, affine transformations of independent normals are normal:

In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-cmgj2l
Out[2]=2

Normal distribution is closed under addition:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-pbziph
Out[1]=1

The normal distribution is symmetric about its mean:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-iv7ddw
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-3z34n1
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfuuto891sq-nea2ei
Out[3]=3

Parameter mixture of a normal distribution with a normal distribution is again a normal distribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-5lloa7
Out[1]=1

Relationships to other distributions:

Normal (SN) JohnsonDistribution is a normal distribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-vo26um
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-3nfssv
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfuuto891sq-zi6gr9
Out[3]=3

StudentTDistribution goes to a normal distribution as goes to :

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-kkestu
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-pq23o1
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfuuto891sq-otyrn7
Out[3]=3

Normal distribution is a transformation of LogNormalDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-s7w87a
Out[1]=1

The inverse transformation of normal distribution yields LogNormalDistribution:

In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-qgrnvb
Out[2]=2

HalfNormalDistribution is a truncated normal distribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-vi0
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-tsy
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfuuto891sq-91cjn
Out[3]=3

The normal and half-normal distributions:

In[4]:=4
✖
https://wolfram.com/xid/0gfuuto891sq-c8nxb
Out[4]=4

HalfNormalDistribution is a transformation of normal distribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-wkkkpw
Out[1]=1

HalfNormalDistribution is a transformation of normal distribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-hh7vqv
Out[1]=1

NormalDistribution is a special case of ExponentialPowerDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-ko2nr5
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-vk55n2
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfuuto891sq-ryqqd9
Out[3]=3

Normal distribution is a special case of SkewNormalDistribution with shape parameter :

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-mufnhu
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-wrpjmx
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfuuto891sq-wd84a4
Out[3]=3

SkewNormalDistribution is a transformation of normal distribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-gqmtvb
Out[1]=1

Sum of squares of standard normally distributed variables follows ChiSquareDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-0xs75z
Out[1]=1

Sum of squares of normally distributed variables has NoncentralChiSquareDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-9rxmu4
Out[1]=1

The norm of standard normally distributed variables follows ChiDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-f9ow9o
Out[1]=1

The norm of three standard normal variables has MaxwellDistribution, a case of ChiDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-i8678x
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-bw00ec
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfuuto891sq-x5hpe6
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0gfuuto891sq-z80cqt
Out[4]=4

The norm of two standard normally distributed variables follows RayleighDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-zxvaib
Out[1]=1

The norm of two normally distributed variables follows RiceDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-kx2r91
In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-8i0ud9
Out[2]=2

NormalDistribution is the limiting case of HyperbolicDistribution of for and :

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-b1jazy
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-mu1v5w
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfuuto891sq-opm24h
Out[3]=3

If , , , and are independent and normal, then has LaplaceDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-pnylb3
In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-lzm43l
Out[2]=2

Confirm via equality of CharacteristicFunction:

In[3]:=3
✖
https://wolfram.com/xid/0gfuuto891sq-l5ns7n
Out[3]=3

If , , , and are independent and normal, then has LaplaceDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-s8qvl9
In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-flzpbq
Out[2]=2

Confirm via equality of CharacteristicFunction:

In[3]:=3
✖
https://wolfram.com/xid/0gfuuto891sq-bsdm8x
Out[3]=3

Ratio of two normally distributed variables has CauchyDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-161eqt
Out[1]=1

Square of a normally distributed variable is a special case of GammaDistribution, and also of ChiSquareDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-5maj9x
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-42avq
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfuuto891sq-vta6qr
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0gfuuto891sq-xh28fy
Out[4]=4

LaplaceDistribution is a parameter mixture of a normal distribution with RayleighDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-js1h2z
Out[1]=1

StudentTDistribution is a parameter mixture of a normal distribution with GammaDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-r3i4sw
Out[1]=1

LevyDistribution is a transformation of a normal distribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-k9mpt
Out[1]=1

With mean and scale:

In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-eunp24
Out[2]=2

Normal distribution is a special case of type 3 PearsonDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-xnrrmj
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-j864on
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfuuto891sq-et4g0n
Out[3]=3

Normal distribution is a StableDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-xkrde7
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-49eymo
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfuuto891sq-nuxytu
Out[3]=3

Normal distribution is the marginal distribution of BinormalDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-3vqu8o
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-u5we3x
Out[2]=2

Normal distribution is the marginal distribution of MultinormalDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-48ghft
In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-9ebrh8
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfuuto891sq-j7w5h7
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0gfuuto891sq-laf19b
Out[4]=4

NormalDistribution can be obtained from MultinormalDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-e76tlq
Out[1]=1

StudentTDistribution can be obtained from NormalDistribution and ChiSquareDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-uzjzzd
In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-ftmbps
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0gfuuto891sq-7izxbh
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0gfuuto891sq-vlmzkx
Out[4]=4

NoncentralStudentTDistribution can be obtained from NormalDistribution and ChiSquareDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-6ianfa
In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-x9yblq
Out[2]=2

VarianceGammaDistribution can be obtained from GammaDistribution and normal distribution:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-oceirn
In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-cmjxr6
Out[2]=2

Rewrite to simplify:

In[3]:=3
✖
https://wolfram.com/xid/0gfuuto891sq-csdybi
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0gfuuto891sq-z6wqfl
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0gfuuto891sq-hrbojx
Out[5]=5

Possible Issues  (2)Common pitfalls and unexpected behavior

NormalDistribution is not defined when μ is not a real number:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-n2l
Out[1]=1

NormalDistribution is not defined when σ is not a positive real number:

In[2]:=2
✖
https://wolfram.com/xid/0gfuuto891sq-ef2
Out[2]=2

Substitution of invalid parameters into symbolic outputs gives results that are not meaningful:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-cbt
Out[1]=1

Neat Examples  (1)Surprising or curious use cases

PDFs for different σ values with CDF contours:

In[1]:=1
✖
https://wolfram.com/xid/0gfuuto891sq-io0104
In[4]:=4
✖
https://wolfram.com/xid/0gfuuto891sq-ceikus
Out[4]=4
Wolfram Research (2007), NormalDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/NormalDistribution.html (updated 2016).
✖
Wolfram Research (2007), NormalDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/NormalDistribution.html (updated 2016).

Text

Wolfram Research (2007), NormalDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/NormalDistribution.html (updated 2016).

✖
Wolfram Research (2007), NormalDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/NormalDistribution.html (updated 2016).

CMS

Wolfram Language. 2007. "NormalDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/NormalDistribution.html.

✖
Wolfram Language. 2007. "NormalDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/NormalDistribution.html.

APA

Wolfram Language. (2007). NormalDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/NormalDistribution.html

✖
Wolfram Language. (2007). NormalDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/NormalDistribution.html

BibTeX

@misc{reference.wolfram_2025_normaldistribution, author="Wolfram Research", title="{NormalDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/NormalDistribution.html}", note=[Accessed: 29-March-2025 ]}

✖
@misc{reference.wolfram_2025_normaldistribution, author="Wolfram Research", title="{NormalDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/NormalDistribution.html}", note=[Accessed: 29-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_normaldistribution, organization={Wolfram Research}, title={NormalDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/NormalDistribution.html}, note=[Accessed: 29-March-2025 ]}

✖
@online{reference.wolfram_2025_normaldistribution, organization={Wolfram Research}, title={NormalDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/NormalDistribution.html}, note=[Accessed: 29-March-2025 ]}