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Encyclopedia > Normal distribution
Normal
Probability density function
Probability density function for the normal distribution
The green line is the standard normal distribution
Cumulative distribution function
Cumulative distribution function for the normal distribution
Colors match the image above
Parameters μ location (real)
σ2 > 0 squared scale (real)
Support x inmathbb{R}!
Probability density function (pdf) frac1{sigmasqrt{2pi}}; expleft(-frac{left(x-muright)^2}{2sigma^2} right) !
Cumulative distribution function (cdf) frac12 left(1+mathrm{erf},frac{x-mu}{sigmasqrt2}right) !
Mean μ
Median μ
Mode μ
Variance σ2
Skewness 0
Excess kurtosis 0
Entropy lnleft(sigmasqrt{2,pi,e}right)!
Moment-generating function (mgf) M_X(t)= expleft(mu,t+frac{sigma^2 t^2}{2}right)
Characteristic function chi_X(t)=expleft(mu,i,t-frac{sigma^2 t^2}{2}right)

The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. Each member of the family may be defined by two parameters, location and scale: the mean ("average", μ) and variance ("variability", σ2), respectively. The standard normal distribution is the normal distribution with a mean of zero and a variance of one (the green curves in the plots to the right). Carl Friedrich Gauss became associated with this set of distributions when he analyzed astronomical data using them [1], and defined the equation of its probability density function. It is often called the bell curve because the graph of its probability density resembles a bell. Download high resolution version (1300x975, 135 KB) Wikipedia does not have an article with this exact name. ... Download high resolution version (1300x975, 135 KB) Wikipedia does not have an article with this exact name. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In statistics, if a family of probabiblity densities parametrized by a parameter s is of the form fs(x) = f(sx)/s then s is called a scale parameter, since its value determines the scale of the probability distribution. ... In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... This article is about the statistical concept. ... In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ... This article is about mathematics. ... Example of experimental data with non-zero skewness (gravitropic response of wheat coleoptiles, 1,790) In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... The far red light has no effect on the average speed of the gravitropic reaction in wheat coleoptiles, but it changes kurtosis from platykurtic to leptokurtic (-0. ... Claude Shannon In information theory, the Shannon entropy or information entropy is a measure of the uncertainty associated with a random variable. ... In probability theory and statistics, the moment-generating function of a random variable X is wherever this expectation exists. ... In probability theory, the characteristic function of any random variable completely defines its probability distribution. ... By one convention, a probability distribution is called continuous if its cumulative distribution function is continuous. ... In statistics, mean has two related meanings: the arithmetic mean (and is distinguished from the geometric mean or harmonic mean). ... This article is about mathematics. ... In statistics, mean has two related meanings: the arithmetic mean (and is distinguished from the geometric mean or harmonic mean). ... This article is about mathematics. ... Johann Carl Friedrich Gauss or Gauß ( ; Latin: ) (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ... A bell is a simple sound-making device. ...


The importance of the normal distribution as a model of quantitative phenomena in the natural and behavioral sciences is due to the central limit theorem. Many psychological measurements and physical phenomena (like noise) can be approximated well by the normal distribution. While the mechanisms underlying these phenomena are often unknown, the use of the normal model can be theoretically justified by assuming that many small, independent effects are additively contributing to each observation. The Michelson–Morley experiment was used to disprove that light propagated through a luminiferous aether. ... Behavioural sciences (or Behavioral science) is a term that encompasses all the disciplines that explores the behaviour and strategies within and between organisms in the natural world. ... A central limit theorem is any of a set of weak-convergence results in probability theory. ... Psychological science redirects here. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... This article is about noise as in sound. ...


The normal distribution also arises in many areas of statistics. For example, the sampling distribution of the sample mean is approximately normal, even if the distribution of the population from which the sample is taken is not normal. In addition, the normal distribution maximizes information entropy among all distributions with known mean and variance, which makes it the natural choice of underlying distribution for data summarized in terms of sample mean and variance. The normal distribution is the most widely used family of distributions in statistics and many statistical tests are based on the assumption of normality. In probability theory, normal distributions arise as the limiting distributions of several continuous and discrete families of distributions. This article is about the field of statistics. ... In statistics, a sampling distribution is the probability distribution, under repeated sampling of the population, of a given statistic (a numerical quantity calculated from the data values in a sample). ... In mathematics and statistics, the arithmetic mean of a set of numbers is the sum of all the members of the set divided by the number of items in the set. ... Claude Shannon In information theory, the Shannon entropy or information entropy is a measure of the uncertainty associated with a random variable. ... Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... In probability theory, there exist several different notions of convergence of random variables. ... In mathematics, a random variable is discrete if its probability distribution is discrete; a discrete probability distribution is one that is fully characterized by a probability mass function. ...

Contents

History

The normal distribution was first introduced by Abraham de Moivre in an article in 1733, which was reprinted in the second edition of his The Doctrine of Chances, 1738 in the context of approximating certain binomial distributions for large n. His result was extended by Laplace in his book Analytical Theory of Probabilities (1812), and is now called the theorem of de Moivre-Laplace. Abraham de Moivre. ... Events February 12 - British colonist James Oglethorpe founds Savannah, Georgia. ... The Doctrine of Chances is a book on probability theory by 18th-century French mathematician Abraham de Moivre, published in 1733. ... Events February 4 - Court Jew Joseph Suss Oppenheimer is executed in Württenberg April 15 - Premiere in London of Serse, an Italian opera by George Frideric Handel. ... In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. ... Pierre-Simon Laplace Pierre-Simon Laplace (March 23, 1749 – March 5, 1827) was a French mathematician and astronomer, the discoverer of the Laplace transform and Laplaces equation. ... For the overture by Tchaikovsky, see 1812 Overture; For the wars, see War of 1812 (USA - United Kingdom) or Patriotic War of 1812 (France - Russia) For the Siberia Airlines plane crashed over the Black Sea on October 4, 2001, see Siberia Airlines Flight 1812 1812 was a leap year starting... In probability theory, the theorem of de Moivre-Laplace is a special case of the central limit theorem. ...


Laplace used the normal distribution in the analysis of errors of experiments. The important method of least squares was introduced by Legendre in 1805. Gauss, who claimed to have used the method since 1794, justified it rigorously in 1809 by assuming a normal distribution of the errors. In statistics, the concepts of error and residual are easily confused with each other. ... Least squares is a mathematical optimization technique that attempts to find a best fit to a set of data by attempting to minimize the sum of the squares of the differences (called residuals) between the fitted function and the data. ... Adrien-Marie Legendre (September 18, 1752–January 10, 1833) was a French mathematician. ... 1805 was a common year starting on Tuesday (see link for calendar). ... Johann Carl Friedrich Gauss or Gauß ( ; Latin: ) (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... 1794 was a common year starting on Wednesday (see link for calendar). ... Year 1809 (MDCCCIX) was a common year starting on Sunday (link will display the full calendar). ...


The name "bell curve" goes back to Jouffret who first used the term "bell surface" in 1872 for a bivariate normal with independent components. The name "normal distribution" was coined independently by Charles S. Peirce, Francis Galton and Wilhelm Lexis around 1875. This terminology unfortunately encourages the fallacy that many or all other probability distributions are not "normal". (See the discussion of "occurrence" below.) The Jouffret was a French automobile manufactured between 1920 and 1926. ... Year 1872 (MDCCCLXXII) was a leap year starting on Monday (link will display the full calendar) of the Gregorian Calendar (or a leap year starting on Saturday of the 12-day slower Julian calendar). ... In probability theory and statistics, a multivariate normal distribution, also sometimes called a multivariate Gaussian distribution, is a specific probability distribution, which can be thought of as a generalization to higher dimensions of the one-dimensional normal distribution (also called a Gaussian distribution). ... Charles Sanders Peirce Charles Sanders Peirce (September 10, 1839 – April 19, 1914) was an American logician, philosopher, scientist, and mathematician. ... This article does not cite any references or sources. ... Wilhelm Lexis (1837 – 1914) was an eminent German economist and social scientist and a founder of the interdisciplinary study of insurance. ... 1875 (MDCCCLXXV) was a common year starting on Friday (see link for calendar). ...


Characterization

There are various ways to characterize a probability distribution. The most visual is the probability density function (PDF); the PDF of the normal distribution is plotted at the beginning of this article. Equivalent ways are the cumulative distribution function, the moments, the cumulants, the characteristic function, the moment-generating function, the cumulant-generating function, and Maxwell's theorem. See probability distribution for a discussion. In the jargon of mathematics, the statement that Property P characterizes object X means, not simply that X has property P, but that X is the only thing that has property P. It is also common to find statements such as Property Q characterises Y up to isomorphism. The first... In probability theory, every random variable may be attributed to a function defined on a state space equipped with a probability distribution that assigns a probability to every subset (more precisely every measurable subset) of its state space in such a way that the probability axioms are satisfied. ... In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... -1... // Cumulants of probability distributions In probability theory and statistics, the cumulants κn of the probability distribution of a random variable X are given by In other words, κn/n! is the nth coefficient in the power series representation of the logarithm of the moment-generating function. ... In probability theory, the characteristic function of any random variable completely defines its probability distribution. ... In probability theory and statistics, the moment-generating function of a random variable X is wherever this expectation exists. ... In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ... In probability theory, Maxwells theorem, named in honor of James Clerk Maxwell, states that if the probability distribution of a vector-valued random variable X = ( X1, ..., Xn )T is the same as the distribution of GX for every n×n orthogonal matrix G and the components are independent, then... In probability theory, every random variable may be attributed to a function defined on a state space equipped with a probability distribution that assigns a probability to every subset (more precisely every measurable subset) of its state space in such a way that the probability axioms are satisfied. ...


To indicate that a real-valued random variable X is normally distributed with mean μ and variance σ² ≥ 0, we write In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ...

X sim N(mu, sigma^2).,!

While it is certainly useful for certain limit theorems (e.g. asymptotic normality of estimators) and for the theory of Gaussian processes to consider the probability distribution concentrated at μ (see Dirac measure) as a normal distribution with mean μ and variance σ² = 0, this degenerate case is often excluded from the considerations because no density with respect to the Lebesgue measure exists. In statistics, an estimator is a function of the observable sample data that is used to estimate an unknown population parameter; an estimate is the result from the actual application of the function to a particular set of data. ... A Gaussian process is a stochastic process {Xt}t ∈T such that every finite linear combination of the Xt (or, more generally, any linear functional of the sample function Xt) is normally distributed. ... In mathematics, a Dirac measure is a measure δx on a set X that gives a given element x measure 1, so that δx({x}) = 1 and in general δx(Y) = 0 for any subset Y of X not containing x, δx(Z) = 1 for any... In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space. ...


The normal distribution may also be parameterized using a precision parameter τ, defined as the reciprocal of σ². This parameterization has an advantage in numerical applications where σ² is very close to zero and is more convenient to work with in analysis as τ is a natural parameter of the normal distribution. In Wikipedia, precision has the following meanings: In engineering, science, industry and statistics, precision characterises the degree of mutual agreement among a series of individual measurements, values, or results - see accuracy and precision. ... In probability and statistics, an exponential family is any class of probability distributions having a certain form. ...


Probability density function

The continuous probability density function of the normal distribution is the Gaussian function In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ... Gaussian curves parametrised by expected value and variance (see normal distribution) A Gaussian function (named after Carl Friedrich Gauss) is a function of the form: for some real constants a > 0, b, and c. ...

varphi_{mu,sigma^2}(x) = frac{1}{sigmasqrt{2pi}} ,expbiggl( -frac{(x- mu)^2}{2sigma^2}biggr) = frac{1}{sigma} varphileft(frac{x - mu}{sigma}right),quad xinmathbb{R},

where σ > 0 is the standard deviation, the real parameter μ is the expected value, and In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are...

varphi(x)=varphi_{0,1}(x)=frac{1}{sqrt{2pi,}} , e^{-frac{x^2}{2}},quad xinmathbb{R},

is the density function of the "standard" normal distribution, i.e., the normal distribution with μ = 0 and σ = 1. To verify that the integral of varphi_{mu,sigma^2} over the real line is indeed equal to one, see Gaussian integral. This article is about the concept of integrals in calculus. ... In mathematics, the real line is simply the set of real numbers. ... The integral of any Gaussian function (named after Carl Friedrich Gauss) is quickly reducible to the Gaussian integral This integral cannot be computed by elementary means since the function has no simple antiderivative. ...


As a Gaussian function with the denominator of the exponent equal to 2, the standard normal density function scriptstylevarphi is an eigenfunction of the Fourier transform. In mathematics, an eigenfunction of a linear operator A defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ...


Some notable qualities of the probability density function:

  • The density function is symmetric about its mean value μ.
  • The mean μ is also its mode and median.
  • The inflection points of the curve occur at one standard deviation away from the mean, i.e. at μ − σ and μ + σ.

In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ... This article is about the statistical concept. ... Plot of y = x3 with inflection point of (0,0). ...

Cumulative distribution function

The cumulative distribution function (cdf) of a probability distribution, evaluated at a number (lower-case) x, is the probability of the event that a random variable (capital) X with that distribution is less than or equal to x. The cumulative distribution function of the normal distribution is expressed in terms of the density function as follows: In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... In probability theory, every random variable may be attributed to a function defined on a state space equipped with a probability distribution that assigns a probability to every subset (more precisely every measurable subset) of its state space in such a way that the probability axioms are satisfied. ... In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ...

begin{align} Phi_{mu,sigma^2}(x) &{}=int_{-infty}^xvarphi_{mu,sigma^2}(u),du &{}=frac{1}{sigmasqrt{2pi}} int_{-infty}^x exp Bigl( -frac{(u - mu)^2}{2sigma^2} Bigr), du &{}= PhiBigl(frac{x-mu}{sigma}Bigr),quad xinmathbb{R}, end{align}

where the standard normal cdf, Φ, is just the general cdf evaluated with μ = 0 and σ = 1:

Phi(x) = Phi_{0,1}(x) = frac{1}{sqrt{2pi}} int_{-infty}^x expBigl(-frac{u^2}{2}Bigr) , du, quad xinmathbb{R}.

The standard normal cdf can be expressed in terms of a special function called the error function, as In mathematics, several functions are important enough to deserve their own name. ... Plot of the error function In mathematics, the error function (also called the Gauss error function) is a non-elementary function which occurs in probability, statistics and partial differential equations. ...

Phi(x) =frac{1}{2} Bigl[ 1 + operatorname{erf} Bigl( frac{x}{sqrt{2}} Bigr) Bigr], quad xinmathbb{R},

and the cdf itself can hence be expressed as

Phi_{mu,sigma^2}(x) =frac{1}{2} Bigl[ 1 + operatorname{erf} Bigl( frac{x-mu}{sigmasqrt{2}} Bigr) Bigr], quad xinmathbb{R}.

The complement of the standard normal cdf, 1 − Φ(x), is often denoted Q(x), and is sometimes referred to simply as the Q-function, especially in engineering texts.[2][3] This represents the tail probability of the Gaussian distribution. Other definitions of the Q-function, all of which are simple transformations of Φ, are also used occasionally.[4]


The inverse standard normal cumulative distribution function, or quantile function, can be expressed in terms of the inverse error function: This article or section does not cite any references or sources. ...

Phi^{-1}(p) = sqrt2 ;operatorname{erf}^{-1} (2p - 1), quad pin(0,1),

and the inverse cumulative distribution function can hence be expressed as

Phi_{mu,sigma^2}^{-1}(p) = mu + sigmaPhi^{-1}(p) = mu + sigmasqrt2 ; operatorname{erf}^{-1}(2p - 1), quad pin(0,1).

This quantile function is sometimes called the probit function. There is no elementary primitive for the probit function. This is not to say merely that none is known, but rather that the non-existence of such an elementary primitive has been proved. Several accurate methods exist for approximating the quantile function for the normal distribution - see quantile function for a discussion and references. In probability theory and statistics the probit function is the inverse cumulative distribution function, or quantile function of the normal distribution. ... In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i. ... This article or section does not cite any references or sources. ...


The values Φ(x) may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and continued fractions. Numerical Integration with the Monte Carlo method: Nodes are random equally distributed. ... Series expansion redirects here. ... In mathematics and applications, particularly the analysis of algorithms, asymptotic analysis is a method of classifying limiting behaviour, by concentrating on some trend. ... In complex analysis, the continued fraction of Gauss is a particular continued fraction derived from the hypergeometric functions. ...


Strict lower and upper bounds for the cdf

For large x the standard normal cdf scriptstylePhi(x) is close to 1 and scriptstylePhi(-x),{=},1,{-},Phi(x) is close to 0. The elementary bounds

frac{x}{1+x^2}varphi(x)<1-Phi(x)<frac{varphi(x)}{x}, qquad x>0,

in terms of the density scriptstylevarphi are useful.


Using the substitution v = u²/2, the upper bound is derived as follows: In calculus, the substitution rule is a tool for finding antiderivatives and integrals. ...

begin{align} 1-Phi(x) &=int_x^inftyvarphi(u),du &<int_x^inftyfrac uxvarphi(u),du =int_{x^2/2}^inftyfrac{e^{-v}}{xsqrt{2pi}},dv =-biggl.frac{e^{-v}}{xsqrt{2pi}}biggr|_{x^2/2}^infty =frac{varphi(x)}{x}. end{align}

Similarly, using scriptstylevarphi'(u),{=},-u,varphi(u) and the quotient rule, In calculus, the quotient rule is a method of finding the derivative of a function that is the quotient of two other functions for which derivatives exist. ...

begin{align} Bigl(1+frac1{x^2}Bigr)(1-Phi(x)) &=int_x^infty Bigl(1+frac1{x^2}Bigr)varphi(u),du &>int_x^infty Bigl(1+frac1{u^2}Bigr)varphi(u),du =-biggl.frac{varphi(u)}ubiggr|_x^infty =frac{varphi(x)}x. end{align}

Solving for scriptstyle 1,{-},Phi(x), provides the lower bound.


Generating functions

Moment generating function

The moment generating function is defined as the expected value of exp(tX). For a normal distribution, the moment generating function is In probability theory and statistics, the moment-generating function of a random variable X is The moment-generating function generates the moments of the probability distribution, as follows: If X has a continuous probability density function f(x) then the moment generating function is given by where is the ith... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are...

begin{align} M_X(t) & {} = mathrm{E} left[ exp{(tX)} right] & {} = int_{-infty}^{infty} frac{1}{sigma sqrt{2pi} } exp{left( -frac{(x - mu)^2}{2 sigma^2} right)} exp{(tx)} , dx & {} = exp{ left( mu t + frac{sigma^2 t^2}{2} right)} end{align}

as can be seen by completing the square in the exponent. Completing the square is an algebra technique, also used in many types of calculus. ...


Cumulant generating function

The cumulant generating function is the logarithm of the moment generating function: g(t) = μt + σ²t²/2. Since this is a quadratic polynomial in t, only the first two cumulants are nonzero. // Cumulants of probability distributions In probability theory and statistics, the cumulants κn of the probability distribution of a random variable X are given by In other words, κn/n! is the nth coefficient in the power series representation of the logarithm of the moment-generating function. ...


Characteristic function

The characteristic function is defined as the expected value of exp(itX), where i is the imaginary unit. So the characteristic function is obtained by replacing t with i t in the moment-generating function. In probability theory, the characteristic function of any random variable completely defines its probability distribution. ... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ...


For a normal distribution, the characteristic function is

begin{align} chi_X(t;mu,sigma) &{} = M_X(i t) = mathrm{E} left[ exp(i t X) right] &{}= int_{-infty}^{infty} frac{1}{sigma sqrt{2pi}} exp left(- frac{(x - mu)^2}{2sigma^2} right) exp(i t x) , dx &{}= exp left( i mu t - frac{sigma^2 t^2}{2} right). end{align}

Properties

Some properties of the normal distribution:

  1. If X sim N(mu, sigma^2) and a and b are real numbers, then a X + b sim N(a mu + b, (a sigma)^2) (see expected value and variance).
  2. If X sim N(mu_X, sigma^2_X) and Y sim N(mu_Y, sigma^2_Y) are independent normal random variables, then:
    • Their sum is normally distributed with U = X + Y sim N(mu_X + mu_Y, sigma^2_X + sigma^2_Y) (proof). Interestingly, the converse holds: if two independent random variables have a normally-distributed sum, then they must be normal themselves — this is known as Cramér's theorem.
    • Their difference is normally distributed with V = X - Y sim N(mu_X - mu_Y, sigma^2_X + sigma^2_Y).
    • If the variances of X and Y are equal, then U and V are independent of each other.
    • The Kullback-Leibler divergence, D_{rm KL}( X | Y ) = { 1 over 2 } left( log left( { sigma^2_Y over sigma^2_X } right) + frac{sigma^2_X}{sigma^2_Y} + frac{left(mu_Y - mu_Xright)^2}{sigma^2_Y} - 1right).
  3. If X sim N(0, sigma^2_X) and Y sim N(0, sigma^2_Y) are independent normal random variables, then:
  4. If X_1, dots, X_n are independent standard normal variables, then X_1^2 + cdots + X_n^2 has a chi-square distribution with n degrees of freedom.

In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... This article is about mathematics. ... In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ... In probability theory, if X and Y are independent random variables that are normally distributed, then X + Y is also normally distributed. ... In mathematics, Cramérs theorem is the result that if X and Y are independent real-valued random variables whose sum X + Y is a normal random variable, then both X and Y must be normal as well. ... In probability theory and information theory, the Kullback-Leibler divergence (or information divergence, or information gain, or relative entropy) is a natural distance measure from a true probability distribution P to an arbitrary probability distribution Q. Typically P represents data, observations, or a precise calculated probability distribution. ... In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: x2 for an arbitrary real or complex number α. The most common and important special case is where α is an integer n, then α is referred... The Cauchy-Lorentz distribution, named after Augustin Cauchy, is a continuous probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and γ is the scale parameter which specifies the half-width at half-maximum (HWHM). ... A ratio distribution (or quotient distribution) is a statistical distribution constructed as the distribution of the ratio of random variables having two other distributions. ... This article is about the mathematics of the chi-square distribution. ...

Standardizing normal random variables

As a consequence of Property 1, it is possible to relate all normal random variables to the standard normal.


If X ~ N(μ,σ2), then

Z = frac{X - mu}{sigma} !

is a standard normal random variable: Z ~ N(0,1). An important consequence is that the cdf of a general normal distribution is therefore

Pr(X le x) = Phi left( frac{x-mu}{sigma} right) = frac{1}{2} left( 1 + operatorname{erf} left( frac{x-mu}{sigmasqrt{2}} right) right) .

Conversely, if Z is a standard normal distribution, Z ~ N(0,1), then

X = σZ + μ

is a normal random variable with mean μ and variance σ2.


The standard normal distribution has been tabulated (usually in the form of value of the cumulative distribution function Φ), and the other normal distributions are the simple transformations, as described above, of the standard one. Therefore, one can use tabulated values of the cdf of the standard normal distribution to find values of the cdf of a general normal distribution.


Moments

Some of the first few moments of the normal distribution are:-1...

Number Raw moment Central moment Cumulant
0 1 1
1 μ 0 μ
2 μ2 + σ2 σ2 σ2
3 μ3 + 3μσ2 0 0
4 μ4 + 6μ2σ2 + 3σ4 3σ4 0
5 μ5 + 10μ3σ2 + 15μσ4 0 0
6 μ6 + 15μ4σ2 + 45μ2σ4 + 15σ6 15σ6 0
7 μ7 + 21μ5σ2 + 105μ3σ4 + 105μσ6 0 0
8 μ8 + 28μ6σ2 + 210μ4σ4 + 420μ2σ6 + 105σ8 105σ8 0

All cumulants of the normal distribution beyond the second are zero. // Cumulants of probability distributions In probability theory and statistics, the cumulants κn of the probability distribution of a random variable X are given by In other words, κn/n! is the nth coefficient in the power series representation of the logarithm of the moment-generating function. ...


Higher central moments (of order 2k with μ = 0) can be obtained using the formula


Eleft[x^{2k}right]=frac{(2k)!}{2^k k!} sigma^{2k}


Generating values for normal random variables

For computer simulations, it is often useful to generate values that have a normal distribution. There are several methods and the most basic is to invert the standard normal cdf. More efficient methods are also known, one such method being the Box-Muller transform. An even faster algorithm is the ziggurat algorithm. Diagram of the Box Muller transform. ... The ziggurat algorithm generates normally-distributed random variables. ...


The Box-Muller algorithm says that, if you have two numbers a and b uniformly distributed on (0, 1], (e.g. the output from a random number generator), then two standard normally distributed random variables are c and d, where: In mathematics, the uniform distributions are simple probability distributions. ... A random number generator is a computational or physical device designed to generate a sequence of elements (usually numbers), such that the sequence can be used as a random one. ...

c = sqrt{- 2 ln a} cdot cos(2 pi b)
d = sqrt{- 2 ln a} cdot sin(2 pi b)

This is because the chi-square distribution with two degrees of freedom (see property 4 above) is an easily-generated exponential random variable.


The central limit theorem

Main article: central limit theorem
Plot of the pdf of a normal distribution with μ = 12 and σ = 3, approximating the pdf of a binomial distribution with n = 48 and p = 1/4
Plot of the pdf of a normal distribution with μ = 12 and σ = 3, approximating the pdf of a binomial distribution with n = 48 and p = 1/4

Under certain conditions (such as being independent and identically-distributed with finite variance), the sum of a large number of random variables is approximately normally distributed — this is the central limit theorem. A central limit theorem is any of a set of weak-convergence results in probability theory. ... Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... In probability theory, a sequence or other collection of random variables is independent and identically distributed (i. ...


The practical importance of the central limit theorem is that the normal cumulative distribution function can be used as an approximation to some other cumulative distribution functions, for example:

  • A binomial distribution with parameters n and p is approximately normal for large n and p not too close to 1 or 0 (some books recommend using this approximation only if np and n(1 − p) are both at least 5; in this case, a continuity correction should be applied).
    The approximating normal distribution has parameters μ = np, σ2 = np(1 − p).
  • A Poisson distribution with parameter λ is approximately normal for large λ.
    The approximating normal distribution has parameters μ = σ2 = λ.

Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution. It is typically the case that such approximations are less accurate in the tails of the distribution. A general upper bound of the approximation error of the cumulative distribution function is given by the Berry–Esséen theorem. In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. ... In probability theory, if a random variable X has a binomial distribution with parameters n and p, i. ... In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate, and are independent of the time since the last event. ... The central limit theorem in probability theory and statistics states that under certain circumstances the sample mean, considered as a random quantity, becomes more normally distributed as the sample size is increased. ...


Infinite divisibility

The normal distributions are infinitely divisible probability distributions: Given a mean μ, a variance σ 2 ≥ 0, and a natural number n, the sum of n independent random variables In probability theory, to say that a probability distribution F on the real line is infinitely divisible means that if X is any random variable whose distribution is F, then for every positive integer n there exist n independent identically distributed random variables X1, ..., Xn whose sum is equal in...

X_1,X_2,dots,X_n sim N(mu/n, sigma^2!/n),

has this specified normal distribution (to verify this, use characteristic functions or convolution and mathematical induction). In probability theory, if X and Y are independent random variables that are normally distributed, then X + Y is also normally distributed. ... Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ...


Stability

The normal distributions are strictly stable probability distributions. In probability theory and statistics, the stability of a family of probability distributions is an important property which basically states that if you have a number of random variates that are in the family, any linear combination of these variates will also be in the family. Here a family of...


Standard deviation and confidence intervals

Dark blue is less than one standard deviation from the mean. For the normal distribution, this accounts for about 68% of the set (dark blue) while two standard deviations from the mean (medium and dark blue) account for about 95% and three standard deviations (light, medium, and dark blue) account for about 99.7%.
Dark blue is less than one standard deviation from the mean. For the normal distribution, this accounts for about 68% of the set (dark blue) while two standard deviations from the mean (medium and dark blue) account for about 95% and three standard deviations (light, medium, and dark blue) account for about 99.7%.

About 68% of values drawn from a normal distribution are within one standard deviation σ > 0 away from the mean μ; about 95% of the values are within two standard deviations and about 99.7% lie within three standard deviations. This is known as the "68-95-99.7 rule" or the "empirical rule." Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ... In statistics, mean has two related meanings: the arithmetic mean (and is distinguished from the geometric mean or harmonic mean). ... ... ...


To be more precise, the area under the bell curve between μ − nσ and μ + nσ in terms of the cumulative normal distribution function is given by

begin{align}&Phi_{mu,sigma^2}(mu+nsigma)-Phi_{mu,sigma^2}(mu-nsigma) &=Phi(n)-Phi(-n)=2Phi(n)-1=mathrm{erf}bigl(n/sqrt{2},bigr),end{align}

where erf is the error function. To 12 decimal places, the values for the 1-, 2-, up to 6-sigma points are: Plot of the error function In mathematics, the error function (also called the Gauss error function) is a non-elementary function which occurs in probability, statistics and partial differential equations. ...

 n,  mathrm{erf}bigl(n/sqrt{2},bigr),
1  0.682689492137 
2 0.954499736104
3 0.997300203937
4 0.999936657516
5 0.999999426697
6 0.999999998027

The next table gives the reverse relation of sigma multiples corresponding to a few often used values for the area under the bell curve. These values are useful to determine (asymptotic) confidence intervals of the specified levels for normally distributed (or asymptotically normal) estimators: In this diagram, the bars represent observation means and the red lines represent the confidence intervals surrounding them. ... In statistics, an estimator is a function of the observable sample data that is used to estimate an unknown population parameter; an estimate is the result from the actual application of the function to a particular set of data. ... In statistics, an estimator is a function of the observable sample data that is used to estimate an unknown population parameter; an estimate is the result from the actual application of the function to a particular set of data. ...

 mathrm{erf}bigl(n/sqrt{2},bigr)  n, 
0.80  1.28155 
0.90 1.64485
0.95 1.95996
0.98 2.32635
0.99 2.57583
0.995 2.80703
0.998 3.09023
0.999 3.29052

where the value on the left of the table is the proportion of values that will fall within a given interval and n is a multiple of the standard deviation that specifies the width of the interval.


Exponential family form

The Normal distribution is a two-parameter exponential family form with natural parameters μ and 1/σ2, and natural statistics X and X2. The canonical form has parameters {mu over sigma^2} and {1 over sigma^2} and sufficient statistics sum x and -{1 over 2} sum x^2 . In probability and statistics, an exponential family is any class of probability distributions having a certain form. ...


Complex Gaussian process

Consider complex Gaussian random variable,

Z=X+iY,

where X and Y are real and independent Gaussian variables with equal variances scriptstyle sigma_r^2,. The pdf of the joint variables is then

frac{1}{2,pi,sigma_r^2} e^{-(x^2+y^2)/(2 sigma_r ^2)}

Because scriptstyle sigma_z, =, sqrt{2}sigma_r, the resulting pdf for the complex Gaussian variable Z is

frac{1}{pi,sigma_z^2} e^{-|z|^2/sigma_z^2}.

Related distributions

In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution. ... This article is about the mathematics of the chi-square distribution. ... This article or section is in need of attention from an expert on the subject. ... The Cauchy-Lorentz distribution, named after Augustin Cauchy, is a continuous probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and γ is the scale parameter which specifies the half-width at half-maximum (HWHM). ... In probability and statistics, the log-normal distribution is the single-tailed probability distribution of any random variable whose logarithm is normally distributed. ... In probability theory, a Lévy skew alpha-stable distribution or just stable distribution, developed by Paul Lévy, is a probability distribution where sums of independent identically distributed random variables have the same distribution as the original. ... In probability and statistics, the truncated normal distribution is the probability distribution of a normally distributed random variable whose value is either bounded below or above (or both). ... In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ... The folded normal distribution is a probability distribution related to the normal distribution. ...

Descriptive and inferential statistics

Scores

Many scores are derived from the normal distribution, including percentile ranks ("percentiles"), normal curve equivalents, stanines, z-scores, and T-scores. Additionally, a number of behavioral statistical procedures are based on the assumption that scores are normally distributed; for example, t-tests and ANOVAs (see below). Bell curve grading assigns relative grades based on a normal distribution of scores. The percentile rank of a score is the percentage of scores in its frequency distribution which are lower. ... Normal curve equivalents (NCEs) compared to Percentile ranks (PRs or percentiles) A normal curve equivalent (NCE), developed by the Department of Education,[1] is a score received on a test based on the percentile rank. ... Stanine (STAndard NINE) is a method of scaling test scores on a nine point standard scale with a mean of 5 and a standard deviation of two. ... Compares the various grading methods in a normal distribution. ... This article is about the field of statistics. ... A t test is any statistical hypothesis test in which the test statistic has a Students t distribution if the null hypothesis is true. ... In statistics, analysis of variance (ANOVA) is a collection of statistical models and their associated procedures which compare means by splitting the overall observed variance into different parts. ... In education, grading on a bell curve is grading a group of examinations first using a numerical point system, then assigning the highest grade an A, regardless of its numerical grade (which can be failing). ...


Normality tests

Main article: normality test

Normality tests check a given set of data for similarity to the normal distribution. The null hypothesis is that the data set is similar to the normal distribution, therefore a sufficiently small P-value indicates non-normal data. In statistics, normality tests are concerned with determining whether or not a random variable is normally distributed. ... In statistics, a null hypothesis is a hypothesis set up to be nullified or refuted in order to support an alternative hypothesis. ... In statistical hypothesis testing, the p-value of a random variable T used as a test statistic is the probability that T will assume a value at least as extreme as the observed value tobserved, given that a null hypothesis being considered is true. ...

In statistics, the Kolmogorov-Smirnov test (often called the K-S test) is used to determine whether two underlying probability distributions differ, or whether an underlying probability distribution differs from a hypothesized distribution, in either case based on finite samples. ... In statistics, the Lilliefors test, named after Hubert Lilliefors, professor of statistics at George Washington University, is an adaptation of the Kolmogorov-Smirnov test. ... The Anderson-Darling test assesses whether known data come from a specified distribution. ... In statistics, the Shapiro-Wilk test tests the null hypothesis that a sample x1, ..., xn came from a normally distributed population. ... In statistics, the rankits of the data points in a data set consisting simply of a list of scalars are expected values of order statistics of the standard normal distribution corresponding to data points in a manner determined by the order in which the data points appear. ... In statistics, the rankits of the data points in a data set consisting simply of a list of scalars are expected values of order statistics of the standard normal distribution corresponding to data points in a manner determined by the order in which the data points appear. ... In statistics, the Jarque-Bera test is a goodness-of-fit measure of departure from normality, based on the sample kurtosis and skewness. ...

Estimation of parameters

Maximum likelihood estimation of parameters

Suppose

X_1,dots,X_n

are independent and each is normally distributed with expectation μ and variance σ² > 0. In the language of statisticians, the observed values of these n random variables make up a "sample of size n from a normally distributed population." It is desired to estimate the "population mean" μ and the "population standard deviation" σ, based on the observed values of this sample. The continuous joint probability density function of these n independent random variables is

begin{align}f(x_1,dots,x_n;mu,sigma) &= prod_{i=1}^n varphi_{mu,sigma^2}(x_i) &=frac1{(sigmasqrt{2pi})^n}prod_{i=1}^n expbiggl(-{1 over 2} Bigl({x_i-mu over sigma}Bigr)^2biggr), quad(x_1,ldots,x_n)inmathbb{R}^n. end{align}

As a function of μ and σ, the likelihood function based on the observations X1, ..., Xn is Look up likelihood in Wiktionary, the free dictionary. ...

L(mu,sigma) = frac C{sigma^n} expleft(-{sum_{i=1}^n (X_i-mu)^2 over 2sigma^2}right), quadmuinmathbb{R}, sigma>0,

with some constant C > 0 (which in general would be even allowed to depend on X1, ..., Xn, but will vanish anyway when partial derivatives of the log-likelihood function with respect to the parameters are computed, see below).


In the method of maximum likelihood, the values of μ and σ that maximize the likelihood function are taken as estimates of the population parameters μ and σ. Maximum likelihood estimation (MLE) is a popular statistical method used to make inferences about parameters of the underlying probability distribution from a given data set. ...


Usually in maximizing a function of two variables, one might consider partial derivatives. But here we will exploit the fact that the value of μ that maximizes the likelihood function with σ fixed does not depend on σ. Therefore, we can find that value of μ, then substitute it for μ in the likelihood function, and finally find the value of σ that maximizes the resulting expression. In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). ...


It is evident that the likelihood function is a decreasing function of the sum

sum_{i=1}^n (X_i-mu)^2. ,!

So we want the value of μ that minimizes this sum. Let

overline{X}_n=(X_1+cdots+X_n)/n

be the "sample mean" based on the n observations. Observe that

begin{align} sum_{i=1}^n (X_i-mu)^2 &=sum_{i=1}^nbigl((X_i-overline{X}_n)+(overline{X}_n-mu)bigr)^2 &=sum_{i=1}^n(X_i-overline{X}_n)^2 + 2(overline{X}_n-mu)underbrace{sum_{i=1}^n (X_i-overline{X}_n)}_{=,0} + sum_{i=1}^n (overline{X}_n-mu)^2 &=sum_{i=1}^n(X_i-overline{X}_n)^2 + n(overline{X}_n-mu)^2. end{align}

Only the last term depends on μ and it is minimized by

widehat{mu}_n=overline{X}_n.

That is the maximum-likelihood estimate of μ based on the n observations X1, ..., Xn. When we substitute that estimate for μ into the likelihood function, we get

L(overline{X}_n,sigma) = frac C{sigma^n} expbiggl(-{sum_{i=1}^n (X_i-overline{X}_n)^2 over 2sigma^2}biggr), quadsigma>0.

It is conventional to denote the "log-likelihood function", i.e., the logarithm of the likelihood function, by a lower-case ell, and we have

ell(overline{X}_n,sigma)=log C-nlogsigma-{sum_{i=1}^n(X_i-overline{X}_n)^2 over 2sigma^2}, quadsigma>0,

and then

begin{align} {partial over partialsigma}ell(overline{X}_n,sigma) &=-{n over sigma} +{sum_{i=1}^n (X_i-overline{X}_n)^2 over sigma^3} &=-{n over sigma^3}biggl(sigma^2-{1 over n}sum_{i=1}^n (X_i-overline{X}_n)^2 biggr), quadsigma>0. end{align}

This derivative is positive, zero, or negative according as σ² is between 0 and

hatsigma_n^2:={1 over n}sum_{i=1}^n(X_i-overline{X}_n)^2,

or equal to that quantity, or greater than that quantity. (If there is just one observation, meaning that n = 1, or if X1 = ... = Xn, which only happens with probability zero, then hatsigma{}_n^2=0 by this formula, reflecting the fact that in these cases the likelihood function is unbounded as σ decreases to zero.)


Consequently this average of squares of residuals is the maximum-likelihood estimate of σ², and its square root is the maximum-likelihood estimate of σ based on the n observations. This estimator hatsigma{}_n^2 is biased, but has a smaller mean squared error than the usual unbiased estimator, which is n/(n − 1) times this estimator. In statistics and optimization, the concepts of error and residual are easily confused with each other. ... In statistics, the difference between an estimators expected value and the true value of the parameter being estimated is called the bias. ... In statistics the mean squared error of an estimator T of an unobservable parameter θ is i. ...


Surprising generalization

The derivation of the maximum-likelihood estimator of the covariance matrix of a multivariate normal distribution is subtle. It involves the spectral theorem and the reason it can be better to view a scalar as the trace of a 1×1 matrix than as a mere scalar. See estimation of covariance matrices. In statistics and probability theory, the covariance matrix is a matrix of covariances between elements of a vector. ... In probability theory and statistics, a multivariate normal distribution, also sometimes called a multivariate Gaussian distribution, is a specific probability distribution, which can be thought of as a generalization to higher dimensions of the one-dimensional normal distribution (also called a Gaussian distribution). ... In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. ... In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector. ... In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... In multivariate statistics, the importance of the Wishart distribution stems in part from the fact that it is the probability distribution of the maximum likelihood estimator of the covariance matrix of a multivariate normal distribution. ...


Unbiased estimation of parameters

The maximum likelihood estimator of the population mean μ from a sample is an unbiased estimator of the mean, as is the variance when the mean of the population is known a priori. However, if we are faced with a sample and have no knowledge of the mean or the variance of the population from which it is drawn, the unbiased estimator of the variance σ2 is: In statistics, a biased estimator is one that for some reason on average over- or underestimates what is being estimated. ...

S^2 = frac{1}{n-1} sum_{i=1}^n (X_i - overline{X})^2.

This "sample variance" follows a Gamma distribution if all Xi are independent and identically-distributed: In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. ... In probability theory, a sequence or other collection of random variables is independent and identically distributed (i. ...

Occurrence

Approximately normal distributions occur in many situations, as a result of the central limit theorem. When there is reason to suspect the presence of a large number of small effects acting additively and independently, it is reasonable to assume that observations will be normal. There are statistical methods to empirically test that assumption, for example the Kolmogorov-Smirnov test. A central limit theorem is any of a set of weak-convergence results in probability theory. ... In statistics, the Kolmogorov-Smirnov test (often called the K-S test) is used to determine whether two underlying probability distributions differ, or whether an underlying probability distribution differs from a hypothesized distribution, in either case based on finite samples. ...


Effects can also act as multiplicative (rather than additive) modifications. In that case, the assumption of normality is not justified, and it is the logarithm of the variable of interest that is normally distributed. The distribution of the directly observed variable is then called log-normal. Look up logarithm in Wiktionary, the free dictionary. ... In probability and statistics, the log-normal distribution is the single-tailed probability distribution of any random variable whose logarithm is normally distributed. ...


Finally, if there is a single external influence which has a large effect on the variable under consideration, the assumption of normality is not justified either. This is true even if, when the external variable is held constant, the resulting marginal distributions are indeed normal. The full distribution will be a superposition of normal variables, which is not in general normal. This is related to the theory of errors (see below).


To summarize, here is a list of situations where approximate normality is sometimes assumed. For a fuller discussion, see below.

  • In counting problems (so the central limit theorem includes a discrete-to-continuum approximation) where reproductive random variables are involved, such as
  • In physiological measurements of biological specimens:
    • The logarithm of measures of size of living tissue (length, height, skin area, weight);
    • The length of inert appendages (hair, claws, nails, teeth) of biological specimens, in the direction of growth; presumably the thickness of tree bark also falls under this category;
    • Other physiological measures may be normally distributed, but there is no reason to expect that a priori;
  • Measurement errors are often assumed to be normally distributed, and any deviation from normality is considered something which should be explained;
  • Financial variables
    • Changes in the logarithm of exchange rates, price indices, and stock market indices; these variables behave like compound interest, not like simple interest, and so are multiplicative;
    • Other financial variables may be normally distributed, but there is no reason to expect that a priori;
  • Light intensity
    • The intensity of laser light is normally distributed;
    • Thermal light has a Bose-Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.

Of relevance to biology and economics is the fact that complex systems tend to display power laws rather than normality. A central limit theorem is any of a set of weak-convergence results in probability theory. ... In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. ... In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate, and are independent of the time since the last event. ... In statistical mechanics, Bose-Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium. ... See Also: Watt In physics, a power law relationship between two scalar quantities x and y is any such that the relationship can be written as where a (the constant of proportionality) and k (the exponent of the power law) are constants. ...


Photon counting

Light intensity from a single source varies with time, as thermal fluctuations can be observed if the light is analyzed at sufficiently high time resolution. The intensity is usually assumed to be normally distributed. Quantum mechanics interprets measurements of light intensity as photon counting. The natural assumption in this setting is the Poisson distribution. When light intensity is integrated over times longer than the coherence time and is large, the Poisson-to-normal limit is appropriate. In modern physics the photon is the elementary particle responsible for electromagnetic phenomena. ... In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate, and are independent of the time since the last event. ...


Measurement errors

Normality is the central assumption of the mathematical theory of errors. Similarly, in statistical model-fitting, an indicator of goodness of fit is that the residuals (as the errors are called in that setting) be independent and normally distributed. The assumption is that any deviation from normality needs to be explained. In that sense, both in model-fitting and in the theory of errors, normality is the only observation that need not be explained, being expected. However, if the original data are not normally distributed (for instance if they follow a Cauchy distribution), then the residuals will also not be normally distributed. This fact is usually ignored in practice. In statistics, the concepts of error and residual are easily confused with each other. ... In statistics and optimization, the concepts of error and residual are easily confused with each other. ... The Cauchy-Lorentz distribution, named after Augustin Cauchy, is a continuous probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and γ is the scale parameter which specifies the half-width at half-maximum (HWHM). ...


Repeated measurements of the same quantity are expected to yield results which are clustered around a particular value. If all major sources of errors have been taken into account, it is assumed that the remaining error must be the result of a large number of very small additive effects, and hence normal. Deviations from normality are interpreted as indications of systematic errors which have not been taken into account. Whether this assumption is valid is debatable.


Physical characteristics of biological specimens

The sizes of full-grown animals is approximately lognormal. The evidence and an explanation based on models of growth was first published in the 1932 book Problems of Relative Growth by Julian Huxley. Sir Julian Sorell Huxley, FRS (June 22, 1887 – February 14, 1975) was a English biologist, author, Humanist and internationalist, known for his popularisations of science in books and lectures. ...


Differences in size due to sexual dimorphism, or other polymorphisms like the worker/soldier/queen division in social insects, further make the distribution of sizes deviate from lognormality.


The assumption that linear size of biological specimens is normal (rather than lognormal) leads to a non-normal distribution of weight (since weight or volume is roughly proportional to the 2nd or 3rd power of length, and Gaussian distributions are only preserved by linear transformations), and conversely assuming that weight is normal leads to non-normal lengths. This is a problem, because there is no a priori reason why one of length, or body mass, and not the other, should be normally distributed. Lognormal distributions, on the other hand, are preserved by powers so the "problem" goes away if lognormality is assumed.


On the other hand, there are some biological measures where normality is assumed, such as blood pressure of adult humans. This is supposed to be normally distributed, but only after separating males and females into different populations (each of which is normally distributed).


Financial variables

Already in 1900 Louis Bachelier suggested to model price changes of stocks by normal distributions. This has been later slightly modified: because of the exponential nature of inflation, financial indicators such as stock values, or commodity prices make good examples of multiplicative behavior. As such, periodic changes in them (for example, yearly changes) should not be expected to be normal, but perhaps lognormal, i.e. there relative returns should be normally distributed. This is still the most commonly used hypothesis in finance, in particular in asset pricing. Corrections to this model seem to be necessary, as has been pointed out for instance by Benoît Mandelbrot, the popularizer of fractals, who observed that the changes in logarithm over short periods (such as a day) are approximated well by distributions that do not have a finite variance, and therefore the central limit theorem does not apply. Rather, the sum of many such changes gives log-Levy distributions. Äž: For the film, see: 1900 (film). ... Louis Jean-Baptiste Alphonse Bachelier (March 11, 1870 - April 28, 1946) was a French mathematician at the turn of the 20th century. ... For other uses, see Stock (disambiguation). ... For other uses, see Stock (disambiguation). ... This article does not cite any references or sources. ... In economics and business, the price is the assigned numerical monetary value of a good, service or asset. ... Finance studies and addresses the ways in which individuals, businesses, and organizations raise, allocate, and use monetary resources over time, taking into account the risks entailed in their projects. ... Valuation is the process of estimating the value of an asset or liability. ... Benoît B. Mandelbrot, PhD, (born November 20, 1924) is a Franco-American mathematician, best known as the father of fractal geometry. Benoît Mandelbrot was born in Poland, but his family moved to France when he was a child; he is a dual French and American citizen and was... A fractal is a geometric object which can be divided into parts, each of which is similar to the original object. ... In probability theory, a Lévy skew alpha-stable distribution or just stable distribution, developed by Paul Lévy, is a probability distribution where sums of independent identically distributed random variables have the same distribution as the original. ...


Distribution in testing and intelligence

Sometimes, the difficulty and number of questions on an IQ test is selected in order to yield normal distributed results. Or else, the raw test scores are converted to IQ values by fitting them to the normal distribution. In either case, it is the deliberate result of test construction or score interpretation that leads to IQ scores being normally distributed for the majority of the population. However, the question whether intelligence itself is normally distributed is more involved, because intelligence is a latent variable, therefore its distribution cannot be observed directly. IQ redirects here. ... Intelligence is the mental capacity to reason, plan, solve problems, think abstractly, comprehend ideas and language, and learn. ... Intelligence is the mental capacity to reason, plan, solve problems, think abstractly, comprehend ideas and language, and learn. ... Latent variables, as opposed to observable variables, are those variables that cannot be directly observed but are rather inferred from other variables that can be observed and directly measured. ...


Numerical approximations of the normal distribution and its CDF

The normal distribution is widely used in scientific and statistical computing. Therefore, it has been implemented in various ways.


The GNU Scientific Library calculates values of the standard normal CDF using piecewise approximations by rational functions. Another approximation method uses third-degree polynomials on intervals [1]. Code using the library and the computed results In computing, GNU Scientific Library (or GSL) is a software library written in the C programming language for numerical calculations in applied mathematics and science. ... In mathematics, a function f(x) of a real number variable x is defined piecewise, if f(x) is given by different expressions on various intervals. ... In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ...


Generation of deviates from the unit normal is normally done using the Box-Muller method of choosing an angle uniformly and a radius exponential and then transforming to (normally distributed) x and y coordinates. If log, cos or sin are expensive then a simple alternative is to simply sum 12 uniform [−1/2, 1/2] deviates. This is equivalent to a twelfth-order polynomial approximation to the normal distribution and is quite usable in many applications. Diagram of the Box Muller transform. ...


A method that is much faster than the Box-Muller transform but which is still exact is the so called Ziggurat algorithm developed by George Marsaglia. In about 97% of all cases it uses only two random numbers, one random integer and one random uniform, one multiplication and an if-test. Only in 3% of the cases where the combination of those two falls outside the "core of the ziggurat" a kind of rejection sampling using logarithms, exponentials and more uniform random numbers has to be employed. The ziggurat algorithm generates normally-distributed random variables. ...


There is also some investigation into the connection between the fast Hadamard transform and the normal distribution since the transform employs just addition and subtraction and by the central limit theorem random numbers from almost any distribution will be transformed into the normal distribution. In this regard a series of Hadamard transforms can be combined with random permutations to turn arbitrary data sets into a normally distributed data. The Hadamard transform (Hadamard transformation, also known as the Walsh-Hadamard transformation) is an example of a generalized class of Fourier transforms. ...


In Microsoft Excel the function NORMSDIST() calculates the cdf of the standard normal distribution, and NORMSINV() calculates its inverse function. Microsoft Excel (full name Microsoft Office Excel) is a spreadsheet application written and distributed by Microsoft for Microsoft Windows and Mac OS. It features calculation and graphing tools which, along with aggressive marketing, have made Excel one of the most popular microcomputer applications to date. ...


Trivia

The Deutsche Mark (DM, DEM) was the official currency of West and, from 1990, unified Germany. ... A £20 Bank of England banknote. ... Johann Carl Friedrich Gauss or Gauß ( ; Latin: ) (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ...

See also

In statistics, the Behrens-Fisher problem is the problem of interval estimation and hypothesis testing concerning the difference between the means of two normally distributed populations when the variances of the two populations are not assumed to be equal, based on two independent samples. ... In statistics, data transformation is carried in order to transform the data and assure that it has a normal distribution (a remedy for outliers, failures of normality, linearity, and homoscedasticity). ... In number theory, the Erdős-Kac theorem, named after Paul Erdős and Mark Kac, states that if ω(x) is the number of distinct prime factors of x, then where is the probability density function of the standard normal distribution, which occurs incessantly in probability theory and statistics. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... Gaussian blur is a widely used effect in graphics software such as Adobe Photoshop, The GIMP, Inkscape, and Paint. ... In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ... Gaussian curves parametrised by expected value and variance (see normal distribution) A Gaussian function (named after Carl Friedrich Gauss) is a function of the form: for some real constants a > 0, b, and c. ... Iannis Xenakis in 1975. ... For other uses, see Music (disambiguation). ... In probability and statistics, the log-normal distribution is the probability distribution of any random variable whose logarithm is normally distributed (the base of the logarithmic function is immaterial in that loga X is normally distributed if and only if logb X is normally distributed). ... In probability theory and statistics, a multivariate normal distribution, also sometimes called a multivariate Gaussian distribution, is a specific probability distribution, which can be thought of as a generalization to higher dimensions of the one-dimensional normal distribution (also called a Gaussian distribution). ... The matrix normal distribution is a probability distribution that is a generalization of the normal distribution. ... In probability theory and statistics, the normal-gamma distribution is a four-parameter family of continuous probability distributions. ... In probability theory, it is almost a cliche to say that uncorrelatedness of two random variables does not entail independence. ... In probability theory and statistics, a multivariate normal distribution, also sometimes called a multivariate Gaussian distribution, is a specific probability distribution, which can be thought of as a generalization to higher dimensions of the one-dimensional normal distribution (also called a Gaussian distribution). ... In probability theory and statistics the probit function is the inverse cumulative distribution function, or quantile function of the normal distribution. ... The sample size of a statistical sample is the number of repeated measurements that constitute it. ... In probability and statistics, the t-distribution or Students t-distribution is a probability distribution that arises in the problem of estimating the mean of a normally distributed population when the sample size is small. ... A Gaussian process is a stochastic process {Xt}t ∈T such that every finite linear combination of the Xt (or, more generally, any linear functional of the sample function Xt) is normally distributed. ... A single realization of a one-dimensional Wiener process A single realization of a three-dimensional Wiener process In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. ... This article may be too technical for most readers to understand. ... In mathematics, the Ornstein-Uhlenbeck process (named after Leonard Salomon Ornstein and George Eugene Uhlenbeck), also known as the mean-reverting process, is a stochastic process rt given by the following stochastic differential equation: where θ, μ and σ are parameters and Wt denotes the Wiener process. ... The probability density function of the inverse Gaussian distribution is given by The Wald distribution is simply another name for the inverse Gaussian distribution. ...

References

  1. ^ Havil, 2003
  2. ^ http://cnx.org/content/m11537/latest/
  3. ^ http://www.eng.tau.ac.il/~jo/academic/Q.pdf
  4. ^ http://mathworld.wolfram.com/NormalDistributionFunction.html

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External links

Probability distributions [ view • talk • edit ]
Univariate Multivariate
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Continuous: Beta • Beta prime • Cauchy • chi-square • Dirac delta function • Coxian • Erlang • exponential • exponential power • F • fading • Fermi-Dirac • Fisher's z • Fisher-Tippett • Gamma • generalized extreme value • generalized hyperbolic • generalized inverse Gaussian • Half-logistic • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square (scaled inverse chi-square) • inverse Gaussian • inverse gamma (scaled inverse gamma) • Kumaraswamy • Landau • Laplace • Lévy • Lévy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speed • Nakagami • normal (Gaussian) • normal-gamma • normal inverse Gaussian • Pareto • Pearson • phase-type • polar • raised cosine • Rayleigh • relativistic Breit-Wigner • Rice • shifted Gompertz • Student's t • triangular • truncated normal • type-1 Gumbel • type-2 Gumbel • uniform • Variance-Gamma • Voigt • von Mises • Weibull • Wigner semicircle • Wilks' lambda Dirichlet • Generalized Dirichlet • inverse-Wishart • Kent • matrix normal • multivariate normal • multivariate Student • von Mises-Fisher • Wigner quasi • Wishart
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v • d • e
Statistics
Descriptive statistics Mean (Arithmetic, Geometric) - Median - Mode - Power - Variance - Standard deviation
Inferential statistics Hypothesis testing - Significance - Null hypothesis/Alternate hypothesis - Error - Z-test - Student's t-test - Maximum likelihood - Standard score/Z score - P-value - Analysis of variance
Survival analysis Survival function - Kaplan-Meier - Logrank test - Failure rate - Proportional hazards models
Probability distributions Normal (bell curve) - Poisson - Bernoulli
Correlation Confounding variable - Pearson product-moment correlation coefficient - Rank correlation (Spearman's rank correlation coefficient, Kendall tau rank correlation coefficient)
Regression analysis Linear regression - Nonlinear regression - Logistic regression

  Results from FactBites:
 
1.3.6.6.1. Normal Distribution (458 words)
is the probability density function of the standard normal distribution.
The location and scale parameters of the normal distribution can be estimated with the sample mean and sample standard deviation, respectively.
The sampling distribution of the mean becomes approximately normal regardless of the distribution of the original variable.
Normal distribution - Wikipedia, the free encyclopedia (3838 words)
The fundamental importance of the normal distribution as a model of quantitative phenomena in the natural and behavioral sciences is due to the central limit theorem (the proof of which requires rather advanced undergraduate mathematics).
The derivation of the maximum-likelihood estimator of the covariance matrix of a multivariate normal distribution is subtle.
In that case, the assumption of normality is not justified, and it is the logarithm of the variable of interest that is normally distributed.
  More results at FactBites »


 

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