Normal distribution

Normal distribution
From Wikipedia, the free encyclopedia
This article is about the univariate normal distribution. For normally distributed vectors,
see Multivariate normal distribution.
Normal
Probability density function
The red curve is the standard normal distribution
Cumulative distribution function
Notation
Parameters μ ∈ R — mean (location)
σ2 > 0 — variance (squared scale)
Support x ∈ R

pdf
CDF
Quantile
Mean μ
Median μ
Mode μ
Variance
Skewness 0
Ex. kurtosis 0
Entropy
MGF
CF
Fisher
information
In probability theory, the normal (or Gaussian) distribution is a very commonly
occurring continuous probability distribution—a function that tells the probability that any real
observation will fall between any two real limits or real numbers, as the curve approaches
zero on either side. Normal distributions are extremely important in statistics and are often
used in the natural and social sciences for real-valued random variables whose distributions
are not known.

The normal distribution is immensely useful because of the central limit theorem, which
states that, under mild conditions, the mean of many random variables independently drawn
from the same distribution is distributed approximately normally, irrespective of the form of
the original distribution: physical quantities that are expected to be the sum of many
independent processes (such as measurement errors) often have a distribution very close to
the normal. Moreover, many results and methods (such as propagation of
uncertainty and least squares parameter fitting) can be derived analytically in explicit form
when the relevant variables are normally distributed.
The Gaussian distribution is sometimes informally called the bell curve. However, many
other distributions are bell-shaped (such as Cauchy's, Student's, and logistic). The
terms Gaussian function and Gaussian bell curve are also ambiguous because they
sometimes refer to multiples of the normal distribution that cannot be directly interpreted in
terms of probabilities.
A normal distribution is
The parameter μ in this definition is the mean or expectation of the distribution (and also
its median and mode). The parameter σis its standard deviation; its variance is
therefore σ 2. A random variable with a Gaussian distribution is said to be normally
distributed and is called a normal deviate.
If μ = 0 and σ = 1, the distribution is called the standard normal distribution or the unit
normal distribution, and a random variable with that distribution is a standard normal
deviate.
The normal distribution is the only absolutely continuousdistribution all of
whose cumulants beyond the first two (i.e., other than the mean and variance) are zero.
It is also the continuous distribution with the maximum entropy for a given mean and
variance.[3][4]
The normal distribution is a subclass of the elliptical distributions. The normal distribution
is symmetric about its mean, and is non-zero over the entire real line. As such it may not
be a suitable model for variables that are inherently positive or strongly skewed, such as
the weight of a person or the price of a share. Such variables may be better described by
other distributions, such as the log-normal distribution or the Pareto distribution.
The value of the normal distribution is practically zero when the value x lies more than a
few standard deviations away from the mean. Therefore, it may not be an appropriate
model when one expects a significant fraction of outliers—values that lie many standard
deviations away from the mean — and least squares and other statistical
inference methods that are optimal for normally distributed variables often become highly

unreliable when applied to such data. In those cases, a more heavy-tailed distribution
should be assumed and the appropriate robust statistical inference methods applied.
The Gaussian distribution belongs to the family of stable distributions which are the
attractors of sums of independent, identically distributed distributions whether or not the
mean or variance is finite. Except for the Gaussian which is a limiting case, all stable
distributions have heavy tails and infinite variance.
Contents
[hide]
 1 Definition
o 1.1 Standard normal distribution
o 1.2 General normal distribution
o 1.3 Notation
o 1.4 Alternative parameterizations
 2 Properties
o 2.1 Symmetries and derivatives
o 2.2 Moments
o 2.3 Fourier transform and characteristic function
o 2.4 Moment and cumulant generating functions
 3 Cumulative distribution function
o 3.1 Standard deviation and tolerance intervals
o 3.2 Quantile function
 4 Zero-variance limit
 5 The central limit theorem
 6 Operations on normal deviates
o 6.1 Infinite divisibility and Cramér's theorem
o 6.2 Bernstein's theorem
 7 Other properties
 8 Related distributions
o 8.1 Operations on a single random variable
o 8.2 Combination of two independent random variables
o 8.3 Combination of two or more independent random variables
o 8.4 Operations on the density function
o 8.5 Extensions
 9 Normality tests
 10 Estimation of parameters
 11 Bayesian analysis of the normal distribution
o 11.1 The sum of two quadratics
 11.1.1 Scalar form
 11.1.2 Vector form
o 11.2 The sum of differences from the mean
o 11.3 With known variance
o 11.4 With known mean
o 11.5 With unknown mean and unknown variance
 12 Occurrence
o 12.1 Exact normality
o 12.2 Approximate normality
o 12.3 Assumed normality
o 12.4 Produced normality
 13 Generating values from normal distribution

 14 Numerical approximations for the normal CDF
 15 History
o 15.1 Development
o 15.2 Naming
 16 See also
 17 Notes
 18 Citations
 19 References
 20 External links
Definition[edit]
Standard normal distribution[edit]
The simplest case of a normal distribution is known as the standard normal distribution.
This is a special case where μ=0 and σ=1, and it is described by this probability density
function:
The factor in this expression ensures that the total area under the curve ϕ(x)
is equal to one.[5] The 1/2 in the exponent ensures that the distribution has unit
variance (and therefore also unit standard deviation). This function is symmetric
around x=0, where it attains its maximum value ; and has inflection
points at +1 and −1.
Authors may differ also on which normal distribution should be called the "standard"
one. Gauss himself defined the standard normal as having variance σ2 = 1/2, that is
Stigler[6] goes even further, defining the standard normal with variance σ2 = 1/2π :
General normal distribution[edit]
Any normal distribution is a version of the standard normal distribution
whose domain has been stretched by a factor σ (the standard deviation) and
then translated by μ (the mean value):
The probability density must be scaled by so that the integral is
still 1.
If Z is a standard normal deviate, then X = Zσ + μ will have a normal
distribution with expected value μ and standard deviation σ. Conversely,

if X is a general normal deviate, then Z = (X − μ)/σ will have a standard
normal distribution.
Every normal distribution is the exponential of a quadratic function:
where a is negative and c is . In this
form, the mean value μ is −b/(2a), and the variance σ2 is −1/(2a).
For the standard normal distribution, a is −1/2, b is zero,
and c is .
Notation[edit]
The standard Gaussian distribution (with zero mean and unit
variance) is often denoted with the Greek letter ϕ (phi).[7] The
alternative form of the Greek phi letter, φ, is also used quite often.
The normal distribution is also often denoted by N(μ, σ2).[8] Thus
when a random variable X is distributed normally with mean μ and
variance σ2, we write
Alternative parameterizations[edit]
Some authors advocate using the precision τ as the parameter
defining the width of the distribution, instead of the deviationσ or
the variance σ2. The precision is normally defined as the
reciprocal of the variance, 1/σ2.[9] The formula for the distribution
then becomes
This choice is claimed to have advantages in numerical
computations when σ is very close to zero and simplify
formulas in some contexts, such as in the Bayesian
inference of variables with multivariate normal distribution.
Occasionally, the precision τ is 1/σ, the reciprocal of the
standard deviation; so that
According to Stigler, this formulation is advantageous
because of a much simpler and easier-to-remember
formula, the fact that the pdf has unit height at zero, and

simple approximate formulas for the quantiles of the
distribution.
Properties[edit]
Symmetries and derivatives[edit]
The normal distribution f(x), with any mean μ and any
positive deviation σ, has the following properties:
 It is symmetric around the point x = μ, which is at
the same time the mode, the median and the mean
of the distribution.[10]
 It is unimodal: its first derivative is positive for x < μ,
negative for x > μ, and zero only at x = μ.
 Its density has two inflection points (where the
second derivative of f is zero and changes sign),
located one standard deviation away from the
mean, namely at x = μ − σ and x = μ + σ.[10]
 Its density is log-concave.[10]
 Its density is infinitely differentiable,
indeed supersmooth of order 2.[11]
 Its second derivative f′′(x) is equal to its derivative
with respect to its variance σ2
Furthermore, the density ϕ of the standard normal
distribution (with μ = 0 and σ = 1) also has the following
properties:
 Its first derivative ϕ′(x) is −xϕ(x).
 Its second derivative ϕ′′(x) is (x2 − 1)ϕ(x)
 More generally, its n-th derivative ϕ(n)(x) is
(−1)nHn(x)ϕ(x), where Hn is the Hermite
polynomial of order n.[12]
 It satisfies the differential equation
or
Moments[edit]
See also: List of integrals of Gaussian
functions

The plain and absolute moments of a
variable X are the expected values
of Xp and |X|p,respectively. If the expected
value μof X is zero, these parameters are
called central moments. Usually we are
interested only in moments with integer
order p.
If X has a normal distribution, these
moments exist and are finite for
any p whose real part is greater than −1.
For any non-negative integer p, the plain
central moments are
Here n!! denotes the double factorial,
that is, the product of every number
from n to 1 that has the same parity
as n.
The central absolute moments coincide
with plain moments for all even orders,
but are nonzero for odd orders. For any
non-negative integer p,
The last formula is valid also for
any non-integer p > −1. When the
mean μ is not zero, the plain and
absolute moments can be
expressed in terms of confluent
hypergeometric
functions 1F1 and U.[citation needed]

These expressions remain
valid even if p is not
integer. See
also generalized Hermite
polynomials.
Order Non-central moment Central moment
1 μ 0
2 μ2
+ σ2
σ 2
3 μ3
+ 3μσ2
0
4 μ4
+ 6μ2
σ2
+ 3σ4
3σ 4
5 μ5
+ 10μ3
σ2
+ 15μσ4
0
6 μ6
+ 15μ4
σ2
+ 45μ2
σ4
+ 15σ6
15σ 6
7 μ7
+ 21μ5
σ2
+ 105μ3
σ4
+ 105μσ6
0
8 μ8
+ 28μ6
σ2
+ 210μ4
σ4
+ 420μ2
σ6
+ 105σ8
105σ 8
Fourier transform
and characteristic
function[edit]
The Fourier transform of a
normal distribution f with
mean μ and
deviation σ is[13]
where i is
the imaginary unit. If
the mean μ is zero, the

first factor is 1, and the
Fourier transform is
also a normal
distribution on
the frequency domain,
with mean 0 and
standard deviation 1/σ.
In particular, the
standard normal
distribution ϕ (with μ=0
and σ=1) is
an eigenfunction of the
Fourier transform.
In probability theory,
the Fourier transform
of the probability
distribution of a real-
valued random
variable X is called
thecharacteristic
function of that
variable, and can be
defined as
the expected
value of eitX, as a
function of the real
variable t(the frequenc
y parameter of the
Fourier transform).
This definition can be
analytically extended
to a complex-value
parameter t.[14]
Moment and
cumulant
generating
functions[edit]
The moment
generating function of
a real random
variable X is the

expected value of etX,
as a function of the
real parametert. For a
normal distribution
with mean μ and
deviation σ, the
moment generating
function exists and is
equal to
The cumulant
generating
function is the
logarithm of the
moment
generating
function, namely
Since this is a
quadratic
polynomial
in t, only the
first
two cumulants
are nonzero,
namely the
mean μ and
the
variance σ2.
Cumulati
ve
distributi
on
function[
edit]
The
cumulative

distribution
function (CDF)
of the
standard
normal
distribution,
usually
denoted with
the capital
Greek
letter (phi),
is the integral
Therefore
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