Normal distribution - Wikipedia. In probability theory, the normal (or Gaussian) distribution is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real- valued random variables whose distributions are not known.

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In its most general form, under some conditions (which include finite variance), it states that averages of samples of observations of random variables independently drawn from independent distributions converge in distribution to the normal, that is, become normally distributed when the number of observations is sufficiently large. Physical quantities that are expected to be the sum of many independent processes (such as measurement errors) often have distributions that are nearly normal.

However, many other distributions are bell- shaped (such as the Cauchy, Student's t, and logistic distributions). Even the term Gaussian bell curve is ambiguous because it may be used to refer to some function defined in terms of the Gaussian function which is not a probability distribution because it is not normalized in that it does not integrate to 1. The probability density of the normal distribution is: f(x. This is a special case when . This function is symmetric around x=0. Install Windows 7 From Hard Drive External there. Gauss defined the standard normal as having variance . Conversely, if X.

Definition Standard normal distribution. The simplest case of a normal distribution is known as the standard normal distribution. This is a special case when = and. Sreekanth Reddy. Sreekanth is the Man behind ReLakhs.com. He is an Independent Certified Financial Planner (CFP), engaged in blogging & property consultancy for the.

This variate is called the standardized form of X. In this form, the mean value is . For the standard normal distribution, a=. The precision is normally defined as the reciprocal of the variance, 1/. It is also the continuous distribution with the maximum entropy for a specified mean and variance.

The normal distribution is symmetric about its mean, and is non- zero over the entire real line. As such it may not be a suitable model for variables that are inherently positive or strongly skewed, such as the weight of a person or the price of a share. Such variables may be better described by other distributions, such as the log- normal distribution or the Pareto distribution. The value of the normal distribution is practically zero when the value x.

Therefore, it may not be an appropriate model when one expects a significant fraction of outliers—values that lie many standard deviations away from the mean—and least squares and other statistical inference methods that are optimal for normally distributed variables often become highly unreliable when applied to such data. In those cases, a more heavy- tailed distribution should be assumed and the appropriate robust statistical inference methods applied.

The Gaussian distribution belongs to the family of stable distributions which are the attractors of sums of independent, identically distributed distributions whether or not the mean or variance is finite. Except for the Gaussian which is a limiting case, all stable distributions have heavy tails and infinite variance. It is one of the few distributions that are stable and that have probability density functions that can be expressed analytically, the others being the Cauchy distribution and the L.

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If the expected value . Usually we are interested only in moments with integer order p. For any non- negative integer p. For any non- negative integer p. When the mean . See also generalized Hermite polynomials. Windows Update Notifier 1 2 2. The expectation of X.

Note that above, density f. If the mean . In particular, the standard normal distribution .

This definition can be analytically extended to a complex- value variable t. For a normal distribution with density f. However, many numerical approximations are known; see below. The two functions are closely related, namely. Other definitions of the Q.

Its antiderivative (indefinite integral) is. As an example, the following Pascal function approximates the CDF: function. CDF(x: extended): extended; varvalue,sum: extended; i: integer; beginsum: =x; value: =x; fori: =1to.

Standard deviation and coverage. This fact is known as the 6. More precisely, the probability that a normal deviate lies in the range between . The quantile function of the standard normal distribution is called the probit function, and can be expressed in terms of the inverse error function. These values are used in hypothesis testing, construction of confidence intervals and Q- Q plots. A normal random variable X.

In particular, the quantile z. These values are useful to determine tolerance interval for sample averages and other statistical estimators with normal (or asymptotically normal) distributions.

Therefore, the normal distribution cannot be defined as an ordinary function when . In the bottom- right graph, smoothed profiles of the previous graphs are rescaled, superimposed and compared with a normal distribution (black curve). The central limit theorem states that under certain (fairly common) conditions, the sum of many random variables will have an approximately normal distribution.

More specifically, where X1. The central limit theorem implies that those statistical parameters will have asymptotically normal distributions.

The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example: 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 for the approximation error in the central limit theorem is given by the Berry–Esseen theorem, improvements of the approximation are given by the Edgeworth expansions. Maximum entropy. This functional can be maximized, subject to the constraints that the distribution is properly normalized and has a specified variance, by using variational calculus. A function with two Lagrange multipliers is defined: L=. This is a special case of the polarization identity. It follows that the normal distribution is stable (with exponent .

This property is called infinite divisibility. The requirement that X and Y should be jointly normal is essential, without it the property does not hold. The dual, expectation parameters for normal distribution are . The same family is flat with respect to the (.

The truncated normal distribution results from rescaling a section of a single density function. Extensions. All these extensions are also called normal or Gaussian laws, so a certain ambiguity in names exists. The multivariate normal distribution describes the Gaussian law in the k- dimensional Euclidean space. A vector X . The variance of X is a k.

The multivariate normal distribution is a special case of the elliptical distributions. As such, its iso- density loci in the k = 2 case are ellipses and in the case of arbitrary k are ellipsoids. Rectified Gaussian distribution a rectified version of normal distribution with all the negative elements reset to 0. Complex normal distribution deals with the complex normal vectors. A complex vector X . The variance- covariance structure of X is described by two matrices: the variance matrix . These can be viewed as elements of some infinite- dimensional Hilbert space H, and thus are the analogues of multivariate normal vectors for the case k = .

A random element h . The variance structure of such Gaussian random element can be described in terms of the linear covariance operator K: H . Several Gaussian processes became popular enough to have their own names.

Gaussian q- distribution is an abstract mathematical construction that represents a . Note that this distribution is different from the Gaussian q- distribution above. A random variable X has a two- piece normal distribution if it has a distributionf.

X(x)=N(. In such case a possible extension would be a richer family of distributions, having more than two parameters and therefore being able to fit the empirical distribution more accurately.