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Gaussian model Let us start with a classical example in which the response signal from a detector is described by a Gaussian error function around the true value with a standard deviation , which is assumed to be exactly known. This model is the best-known among physicists and, indeed, the Gaussian pdf is also known as normal because it is often assumed that errors are normally distributed according to this function. Applying Bayes theorem for continuous variables (see Tab.1), from the likelihood (25)we get for (26)Considering all values of equally likely over a very large interval, we can model the prior with a constant, which simplifies in Eq.(26), yielding (27)Expectation and standard deviation of the posterior distribution are and , respectively. This particular result corresponds to what is often done intuitively in practice. But one has to pay attention to the assumed conditions under which the result is logically valid: Gaussian likelihood and uniform prior. Moreover, we can speak about the probability of true values only in the subjective sense. It is recognized that physicists, and scientists in general, are highly confused about this point (DAgostini 1999a). A noteworthy case of a prior for which the naive inversion gives paradoxical results is when the value of a quantity is constrained to be in the physical region, for example , while falls outside it (or it is at its edge). The simplest prior that cures the problem is a step function , and the result is equivalent to simply renormalizing the pdf in the physical region (this result corresponds to a prescription sometimes used by practitioners with a frequentist background when they encounter this kind of problem). Another interesting case is when the prior knowledge can be modeled with a Gaussian function, for example, describing our knowledge from a previous inference (28)Inserting Eq.(28) into Eq.(26), we get (29)where (30)(31)(32)We can then see that the case corresponds to the limit of a Gaussian prior with very large and finite . The formula for the expected value combining previous knowledge and present experimental information has been written in several ways in Eq.(31). Another enlighting way of writing Eq.(30) is considering and the estimates of at times and , respectively before and after the observation happened at time . Indicating the estimates at different times by , we can rewrite Eq.(30) as (33)(34)(35)where (36)Indeed, we have given Eq.(30) the structure of a Kalman filter (Kalman 1960). The new observation corrects the estimate by a quantity given by the innovation (or residual) times the blending factor (or gain) . For an introduction about Kalman filter and its probabilistic origin, see (Maybeck 1979 and Welch and Bishop 2002). As Eqs.(31)-(35) show, a new experimental information reduces the uncertainty. But this is true as long the previous information and the observation are somewhat consistent. If we are, for several reasons, sceptical about the model which yields the combination rule (31)-(32), we need to remodel the problem and introduce possible systematic errors or underestimations of the quoted standard deviations, as done e.g. in (Press 1997, Dose and von der Linden 1999, DAgostini 1999b, Frhner 2000). Normal distributionFrom Wikipedia, the free encyclopediaJump to: navigation, searchNormalProbability density functionThe red line is the standard normal distributionCumulative distribution functionColors match the image aboveParameters location (real)2 0 squared scale (real)SupportProbability density function (pdf)Cumulative distribution function (cdf)MeanMedianModeVariance2Skewness0Excess kurtosis0EntropyMoment-generating function (mgf)Characteristic functionThe 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, ) and variance (standard deviation squared, 2) respectively. The standard normal distribution is the normal distribution with a mean of zero and a variance of one (the red 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.The importance of the normal distribution as a model of quantitative phenomena in the natural and behavioral sciences is due in part to the central limit theorem. Many measurements, ranging from psychological2 to physical phenomena (in particular, thermal noise) can be approximated, to varying degrees, 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 normal distribution is also important for its relationship to least-squares estimation, one of the simplest and oldest methods of statistical estimation.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.Contentshide 1 History 2 Characterization o 2.1 Probability density function o 2.2 Cumulative distribution function 2.2.1 Strict lower and upper bounds for the cdf o 2.3 Generating functions 2.3.1 Moment generating function 2.3.2 Cumulant generating function 2.3.3 Characteristic function 3 Properties o 3.1 Standardizing normal random variables o 3.2 Moments o 3.3 The central limit theorem o 3.4 Infinite divisibility o 3.5 Stability o 3.6 Standard deviation and confidence intervals o 3.7 Exponential family form 4 Complex Gaussian process 5 Related distributions 6 Descriptive and inferential statistics o 6.1 Scores o 6.2 Normality tests o 6.3 Estimation of parameters 6.3.1 Maximum likelihood estimation of parameters 6.3.1.1 Surprising generalization 6.3.2 Unbiased estimation of parameters 7 Occurrence o 7.1 Photon counting o 7.2 Measurement errors o 7.3 Physical characteristics of biological specimens o 7.4 Financial variables o 7.5 Distribution in testing and intelligence o 7.6 Diffusion equation 8 Use in computational statistics o 8.1 Generating values for normal random variables o 8.2 Numerical approximations of the normal distribution and its cdf 9 See also 10 Notes 11 References 12 External links edit HistoryThe 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.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. The fact the distribution is sometimes called Gaussian is an example of Stiglers Law.The name bell curve goes back to Esprit 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.citation needed Despite this terminology, other probability distributions may be more appropriate in some contexts; see the discussion of occurrence, below.edit CharacterizationThere are various ways to characterize a probability distribution. The most visual is the probability density function (PDF). Equivalent ways are the cumulative distribution function, the moments, the cumulants, the characteristic function, the moment-generating function, the cumulant-generating function, and Maxwells theorem. See probability distribution for a discussion.To indicate that a real-valued random variable X is normally distributed with mean and variance 0, we writeWhile 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 (see Dirac measure) as a normal distribution with mean and variance = 0, this degenerate case is often excluded from the considerations because no density with respect to the Lebesgue measure exists.The normal distribution may also be parameterized using a precision parameter , defined as the reciprocal of . This parameterization has an advantage in numerical applications where is very close to zero and is more convenient to work with in analysis as is a natural parameter of the normal distribution.edit Probability density functionThe continuous probability density function of the normal distribution is the Gaussian functionwhere 0 is the standard deviation, the real parameter is the expected value, andis the density function of the standard normal distribution: i.e., the normal distribution with = 0 and = 1. The integral of over the real line is equal to one as shown in the Gaussian integral article.As a Gaussian function with the denominator of the exponent equal to 2, the standard normal density function is an eigenfunction of the Fourier transform.The probability density function has notable properties including: symmetry about its mean the mode and median both equal the mean the inflection points of the curve occur one standard deviation away from the mean, i.e. at and + . edit Cumulative distribution functionThe 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:The standard normal cdf is just the general cdf evaluated with = 0 and = 1:The standard normal cdf can be expressed in terms of a special function called the error function, asand the cdf itself can hence be expressed asThe complement of the standard normal cdf, 1 (x), is often denoted Q(x), and is sometimes referred to simply as the Q-function, especially in engineering texts.34 This represents the tail probability of the Gaussian distribution. Other definitions of the Q-function, all of which are simple transformations of , are also used occasionally.5The inverse standard normal cumulative distribution function, or quantile function, can be expressed in terms of the inverse error function:and the inverse cumulative distribution function can hence be expressed asThis 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 proven. Several accurate methods exist for approximating the quantile function for the normal distribution - see quantile function for a discussion and references.The values (x) may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and continued fractions.edit Strict lower and upper bounds for the cdfFor large x the standard normal cdf is close to 1 and is close to 0. The elementary boundsin terms of the density are useful.Using the substitution v=u/2, the upper bound is derived as follows:Similarly, using and the quotient rule,Solving for provides the lower bound.edit Generating functionsedit Moment generating functionThe moment generating function is defined as the expected value of exp(tX). For a normal distribution, the moment generating function isas can be seen by completing the square in the exponent.edit Cumulant generating functionThe cumulant generating function is the logarithm of the moment generating function: g(t) = t + t/2. Since this is a quadratic polynomial in t, only the first two cumulants are nonzero.edit Characteristic functionThe 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 it in the moment-generating function.For a normal distribution, the characteristic function isedit PropertiesSome properties of the normal distribution:1. If and a and b are real numbers, then (see expected value and variance). 2. If and are independent normal random variables, then: o Their sum is normally distributed with (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 Cramrs theorem. o Their difference is normally distributed with . o If the variances of X and Y are equal, then U and V are independent of each other. o The Kullback-Leibler divergence, 3. If and are independent normal random variables, then: o Their product XY follows a distribution with density p given by where K0 is a modified Bessel function of the second kind. o Their ratio follows a Cauchy distribution with . Thus the Cauchy distribution is a special kind of ratio distribution. 4. If are independent standard normal variables, then has a chi-square distribution with n degrees of freedom. 5. If are independent standard normal variables, then the sample mean and sample variance are independent. This property characterizes normal distributions (and helps to explain why the F-test is non-robust with respect to non-normality!) edit Standardizing normal random variablesAs a consequence of Property 1, it is possible to relate all normal random variables to the standard normal.If X N(,2), thenis a standard normal random variable: Z N(0,1). An important consequence is that the cdf of a general normal distribution is thereforeConversely, if Z is a standard normal distribution, Z N(0,1), thenX = Z + is a normal random variable with mean and variance 2.The standard normal distribution has been tabulated (usually in the form of value of the cumulative distribution function ), 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.edit MomentsThe first few moments of the normal distribution are:NumberRaw momentCentral momentCumulant0111022 + 22233 + 320044 + 622 + 3434055 + 1032 + 1540066 + 1542 + 4524 + 156156077 + 2152 + 10534 + 10560088 + 2862 + 21044 + 42026 + 105810580All cumulants of the normal distribution beyond the second are zero.Higher central moments (of order 2k with =0) are given by the formulaedit The central limit theoremMain article: central limit theoremPlot of the pdf of a normal distribution with = 12 and = 3, approximating the pdf of a binomial distribution with n = 48 and p = 1/4Under 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.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(1p) are both at least 5; in this case, a continuity correction should be applied).The approximating normal distribution has parameters = np, 2 = np(1p). A Poisson distribution with parameter is approximately normal for large .The approximating normal distribution has parameters = 2 = . 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 BerryEssen theorem.edit Infinite div