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Topic: Sum of normally distributed random variables

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	Normal distribution - Wikipedia, the free encyclopedia
		The normal distribution also arises in many areas of statistics: for example, the sampling distribution of the mean is approximately normal, even if the distribution of the population the sample is taken from is not normal.
		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.
	en.wikipedia.org /wiki/Normal_distribution (4206 words)

	Kenneth L. Kvamme, A Simple Pseudo-Random Normal Deviate Generator (Site not responding. Last check: 2007-10-29)
		The normal distribution follows a bell-shaped curve with observations most probable about a characteristic central value, the arithmetic mean, and probability of occurrence decreasing as one moves in either direction away from the mean to the tail areas (Figures 1b,c).
		The normal is a two-parameter distribution where M is the mean, about which the distribution is symmetric, and V is the variance which determines the width of the distribution (compare Figures 1b,c).
		The variance of a uniform random variable is given by: V=(b-a)2/12, where a is the lower and b is the upper bound of the distribution's** range (Olkin 1980:285).
	www.cast.uark.edu /~kkvamme/ACN37.htm (1011 words)

	Science Fair Projects - Normal distribution
		The normal distribution was first introduced by de Moivre in an article in 1733 (reprinted in the second edition of his The Doctrine of Chances, 1738) in the context of approximating certain binomial distributions for large n.
		While for most practical purposes the distributions of IQ and intelligence (or at least psychometric g) can be seen as the same thing, it is important to distinguish between the two terms when discussing whether they are normally distributed.
		The derivation of the maximum-likelihood estimator of the covariance matrix of a multivariate normal distribution is perhaps surprisingly subtle and elegant.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Normal_distribution (4335 words)

	Central limit theorem - Wikipédia
		be a sequence of random variables which are defined on the same probability space, share the same probability distribution D and are independent.
		But, this limit is just the characteristic function of a standard normal distribution, N(0,1), and the central limit theorem follows from the Lévy continuity theorem, which confirms that the convergence of characteristic functions implies convergence in distribution.
		The density of the sum of two or more independent variables is the convolution of their densities (if these densities exist).
	su.wikipedia.org /wiki/Central_limit_theorem (743 words)

	Central limit theorem - Wikipedia, the free encyclopedia
		Intuitively, they all express the fact that any sum of many independent identically distributed random variables will tend to be distributed according to a particular "attractor distribution".
		Well, the logarithm of a product is simply the sum of the logs of the factors, so the log of a product of random variables tends to have a normal distribution, which makes the product itself have a log-normal distribution.
		There are some theorems which treat the case of sums of non-independent variables, for instance the m-dependent central limit theorem, the martingale central limit theorem and the central limit theorem for mixing processes.
	www.wikipedia.org /wiki/Central_limit_theorem (1318 words)

	Budget cash sample (Site not responding. Last check: 2007-10-29)
		It maximizes a function of random variables subject to the constraint that these variables are maintained at prescribed levels at least with a certain probability.
		Risk represents the size of dispersion of the probability distribution of the input element and it involves situations where it is possible to assign probabilities to the outcomes of a random event.
		In general, the normal distribution is a reasonable assumption in situations in which the central limit theorem applies; for instance, in approximating the distribution of the random variables that are sums of finitely many other random variables.
	budget.998guide.com /budget-cash-sample (1233 words)

	Normal Probability Distribution
		The key to the limiting result lies in the work of the eighteenth century mathematician Abraham de Moivre who showed that the limiting distribution of an additive binomial process is the normal probability distribution.
		In other words, as we move down farther and farther in Pascal’s Triangle, the numbers across a row, divided by the sum of the numbers in the same row, resemble more and more the shape of a normal distribution.
		x is said to be normally distributed with mean m and standard deviation s.
	www.in-the-money.com /presentation/sld085.htm (271 words)

	Differences Between Normal Samples
		A special case of this is with n=2, which can be treated as simply finding the distribution of the differences between two normally distributed populations.
		It's well known that the sum (or difference) of n normally distributed random variables with means u1,u2,..,un and standard deviations s1,s2,..
		The additivity of normal distributions according to equations (0) is so familiar that we often assume it's self-evident, but it's interesting to review how this additivity (which is closely related to the central limit theorem and the special properties of the normal distribution) is actually proven.
	www.mathpages.com /home/kmath046.htm (495 words)

	No Title
		are independent, we have shown that for a random sample from a normal distribution with mean 0 the maximum likelihood estimate of the mean and the maximum likelihood estimate of the variance are independent.
		is normally distributed with mean 0, and the maximum likelihood estimate of the variance is unchanged when the a constant is subtracted from the observations, we have shown that the maximum likelihood estimates of the mean and the variance are independent whenever the population is normally distributed.
		Suppose that we had a population that was known to have a normal distribution, but we knew neither the mean nor the variance of this distribution.
	www.uwm.edu /~ericskey/361material/361F98/L16/index.html (842 words)

	Gaussian distribution: FAQ. D'Errico.
		Even when the right conditions are not met however, the distributions found for many experimentally generated sets of data still tend to have a bell shaped curve that often looks quite like that of a normal.
		Even when a distribution may not be truly normal, it may still be convenient to assume that a normal distribution is a good approximation.
		The standard (mean=0 and variance=1) normal density function is 1 -(1/2)x^2 Z(x) = ---------- e sqrt(2pi) For mean=mu and variance=sigma^2 1 -(1/2)((x-mu)/sigma)^2 Z(x) = ---------------- e sigmasqrt(2pi) The normal* distribution with mean=mu and variance=sigma^2 is customarily referred to as N(mu,sigma^2).
	www.pitt.edu /~wpilib/statfaq/gaussfaq.html (1510 words)

	[No title]
		Y is normally distributed with a mean of -1 and a standard deviation of 1.
		The diameter of a sleeve, denoted D, is normally distributed with a mean of 0.0650 inches and a standard deviation of 0.0002 inches.
		The diameter of a sleeve, denoted d, is normally distributed with a mean of 0.0600 inches and a standard deviation of 0.0002 inches.
	dollar.biz.uiowa.edu /~street/6n216f03/etc/week4_handouts.doc (1150 words)

	Estimating Means from Time Series: Proofs
		This random walk model of the asset return says that each time nature draws out a logarithmic return from the normal distribution we can think of it as being constructed of two parts.
		h is then the sum of normally distributed random variables divided by a constant.
		is not a standard normal random variable, as long as its mean is 0 and its variance is 1, and it is serially uncorrelated.
	www.in-the-money.com /presentation/sld098.htm (267 words)

	Glossary of research economics
		The main advantage of the dummy variable regression is that it provides estimates of he magnitudes of class variation influences on the dependent variables (as well as testing whether or not the classification is meaningful).
		Is the distribution of sums of squares of r standard normal variables.
		This is a one-parameter family of distributions, and the parameter, n, is conventionally labeled the degrees of freedom of the distribution.
	www.econterms.com /econtent.html (14590 words)

	Interpretations of Probability
		Normally, we speak of interpreting a formal system, that is, attaching familiar meanings to the primitive terms in its axioms and theorems, usually with an eye to turning them into true statements about some subject of interest.
		Moreover, various other quantities that have nothing to do with probability do satisfy Kolmogorov's axioms, and thus are interpretations of it in a strict sense: normalized mass, length, area, volume, and indeed anything that falls under the scope of measure theory, the abstract mathematical theory that generalizes such quantities.
		The non-negativity and normalization axioms are largely matters of convention, although it is non-trivial that probability functions take at least the two values 0 and 1, and that they have a maximal value (unlike various other measures, such as length, volume, and so on, which are unbounded).
	plato.stanford.edu /entries/probability-interpret (15163 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		We identify two variants of this approach, a coinductive definition based on final coalgebraic structure in the category of sets, and a direct definition as a final coalgebra in the category of posets.
		The distribution of the faulty pairs themselves however is far from random, with the effect that if the data is so nonrandom as to be just the constant 1, then random calculations started from that constant produce a division error once every few minutes, and these errors will sometimes propagate many more steps.
		Varying the choice of temporal domain D allows for other forms of temporal constraint beyond that available from simple ordering.
	boole.stanford.edu /pub/ABSTRACTS (6481 words)

	Central Limit Theorem
		The central limit theorem states that given a distribution with a mean m and variance s2, the sampling distribution of the mean appraches a normal distribution with a mean and variance/N as N, the sample size, increases.
		If you are are averaging your measurements of a particular observable, your average's distribution may seem to tend toward a normal distribution.
		The random variables Z1, Z2,...., Zn are called the sequence of normalized consecutive sums of the sequence X1, X2,...., Xn.
	software.isixsigma.com /dictionary/Central_Limit_Theorem-177.htm (747 words)

	Publications of the SPACES team (Site not responding. Last check: 2007-10-29)
		It is proved that the mapping is injective, and thus that the parameters of the mixture may be recovered from the values of the moments.
		We consider the requirements for good uniform random number generators, and describe a class of generators whose period is a Mersenne prime or a small multiple of a Mersenne prime.
		Unfortunately, due to the use of various tricks for keeping a good theoretical complexity (sum of squares, infinitesimal deformations, etc.), straightforward implementations of these algorithms are inefficient.
	www-calfor.lip6.fr /~safey/Spaces/publications.html (13078 words)

	Finding Optimal Line Correspondences
		Dividing the squared differences by the variances results in a sum of normally distributed random variables with unary variance -- provided that the match is correct.
		is a sum of squared normal distributed random variables, it is
		In (6) we sum only over the images where the regarded feature exists and add n to compensate for this.
	www.gia.rwth-aachen.de /Forschung/Strassen/Wien/node6.html (785 words)

	Statistics
		It the T-distribution of x with n degrees of freedom; i.e., the probability that n the sum of normally distributed random variables scaled with their mean value and standart deviation are less than x.
		returns the chi^2 distribution; i.e., the distribution of the sum of the squares n normally distributed random variables.
		Note, that the data must be normally distributed for this test to make sense.
	mathsrv.ku-eichstaett.de /MGF/homes/grothman/euler/statistics.html (872 words)

	Mathematics of Volatility
		Changing variables in an SDE is not quite as easy as it is with ordinary differential equations, because any nonlinear transformation alters not only the variable but also the shape and moments of the probability distribution.
		Thus the estimator itself is a random variable, and its theoretical variance can be used to estimate the sampling variation of the statistic in the real world.
		In the case of volatility, the estimator is constructed from a sum of squared normally-distributed random variables.
	www.aetheling.com /MI/Volatility/math.html (2892 words)

	[No title]
		Geometrically, J(w) is proportional to the sum of the distances from the misclassified samples to the decision boundary.
		Since a weighted sum of independent normally distributed random variables is itself normally distributed [e.g., see Mosteller et al.
		Therefore, with the assumption of a linear unit (ADALINE) with normally distributed inputs, the SSE criterion is optimal in the sense of minimizing prediction error.
	neuron.eng.wayne.edu /tarek/MITbook/chap3/3_1.html (6890 words)

	ENGnetBASE: Engineering Handbooks Online
		Mean and Variance for Sum of a Random Number of Random Variables
		Approximations for the Mean and Variance of a Function of a Random Variable
		Distribution of the Sum of Independent Random Variables
	www.engnetbase.com /ejournals/books/book_summary/toc.asp?id=701 (307 words)

	Cochran's theorem (Site not responding. Last check: 2007-10-29)
		is just 1 (it is the of just one linear combination of the normal variables).
		are independent with Chi-squared distribution with n − 1 and 1 degree of respectively.
		Both these distributions are proportional to the but unknown variance andsigma
	www.freeglossary.com /Cochran's_theorem (603 words)

	Notes on Clark Chapter 3 (Econ 308x, Tesfatsion)
		A particular type of ANN supervised training method under which the actual outputs at the ANN output nodes are compared against desired outputs, the resulting overall error (deviation) is calculated, and the various ANN connection weights are then adjusted up or down to decrease this error.
		A "classical" statistician might prefer to work with MSE because of an underlying presumption that the errors are random variables with a known "normal" (bell-shaped) distribution, in which case the MSE (a sum of squared normally distributed random variables) would itself be a random variable with a known distribution.
		As various class members noted, this will presumably depend on the problem or situation we were considering.
	www.econ.iastate.edu /classes/econ308/tesfatsion/clark3a.htm (1662 words)

	Economics Interactive
		The accelerationist view is a variant of natural rate theory that assumes inflationary expectations to be adaptive, so that attempts by policymakers to reduce unemployment below its natural rate requires ever more rapid rates of monetary growth, with the result the inflation accelerates.
		Expectations about the future value of some variable are “adaptive” when extrapolated from (formed as) a weighted average of recent values of that variable.
		Aggregation is the process of summing variables to yield an aggregate variable.
	www.unc.edu /depts/econ/byrns_web/Economicae/EconomicaeA.htm (4071 words)

	[No title]
		As usual the observed value of the dependent variable is a linear function of the fixed value of the independent variable and a stochastic component; that is, EMBED Equation.3 where EMBED Equation.3 are independent normally distributed random variables with mean 0 and variance 1.
		Class Problem: Describe the distribution of EMBED Equation.3 Describe the distribution of EMBED Equation.3 Chapter 2: Checking the Straight Line Fit 2.3 Examining Residuals: The Basic Plots Note that the residuals are not independent (there is a loss of two degrees of freedom).
		That is, the residuals must sum to zero and the residuals must be uncorrelated with the x values.
	www.ams.sunysb.edu /~bere/AMS578/L04_S05.doc (512 words)

	Tax-estimator tax estimator, more information about tax estimator (Site not responding. Last check: 2007-10-29)
		Least squares From Sterwiki 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.
		Regression analysis From Sterwiki Regression analysis is any statistical method where the mean of one or more random variables is predicted conditioned on other (measured) random variables.
		Exponential family From Sterwiki In probability and statistics, the exponential family is an important class of probability distributions.
	www.tax-estimator.us /tax_estimator.html (257 words)

	[No title]
		Positively skewed because the average magnitude of positive deviations is larger than the average magnitude of negative deviations, i.e., the mean return falls above the median.
		They prefer random variables with likely small negative surprises and less likely large positive surprises, to the reverse, likely small good surprises and less likely large bad surprises.
		¡ ãããã$ãTã ó#ª ¨&· Kurtosis for a normal distribution = 3; excess kurtosis is therefore equal to computed kurtosis minus 3.
	www.seattleu.edu /asbe/Directory/syllabi/Yau_Review.ppt (481 words)

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