What is Multivariate normal distribution?
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The variables invovled can be correlated.  Below is a link for the multiviate normal.  Notice that the variance-covariance matrix Σ is not restricted to be diagonal; it can be a general matrix, it just has to be positive semi-definite.  Note that if |Σ| = 0, then the pdf does not exist.  Anyway, the link includes the pdf (in the positive definite case of course), bivariate case, and the conditional distribution, which is also normal.  I have been using this distribution a lot in my research.
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Dimensionality reduction is a strategy, which combines principal components analysis and another technique, such as sliced inverse regression to solve specific biological problems such as to identify linear combinations of genes.  The reference below for an example:
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The definition is given here:\n\nhttp://en.wikipedia.org/wiki/Harmonic_mean\n\nIn words: the harmonic mean of a data set is the inverse of the normal ordinary mean of the inverses of data set.
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The central limit theorem states that given a distribution with a mean μ and variance σ², the sampling distribution of the mean approaches a normal distribution with a mean (μ) and a variance σ²/N as N, the sample size, increases. \n\nThe amazing and counter-intuitive thing about the central limit theorem is that no matter what the shape of the original distribution, the sampling distribution of the mean approaches a normal distribution. \n\nFurthermore, for most distributions, a normal distribution is approached very quickly as N increases. Keep in mind that N is the sample size for each mean and not the number of samples. Remember in a sampling distribution the number of samples is assumed to be infinite.\n\nDEGREE OF FREEDOM:\nThat measure of variability which merely expresses the number of options available within a variable or space. \n\nDegrees of freedom are the equivalent of currency in statistics - you earn a degree of freedom for every data point you collect, and you spend a degree of freedom for each parameter you estimate.
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A guassian pdf is sometimes called a normal pdf, which takes the shape of a bell curve.  There is such a function which is a little messy to fit in here but heres a link so that it looks nice\n\nhttp://science.widener.edu/svb/mathcad/pdf_docs/gaussian.pdf
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It is actually binomial with parameter p=0.5 and n=400. Since 400 is quite a large number, u can approximate this by the Normal distribution with the same mean and variance, which are, np=400 and np(1-p)=100.\n\nX ~ N(200,100)\n\nu wanna test that if 216 is within 95% interval.\n\nVariance=100\nSDev=10\n\n216-200=16,\n16/stdev=1.6\n\nso, 216 is 1.6 stdev from the mean 200. which is within +/-1.96(95% confidence interval. hence, the null hy is accepted, with 95% significance level.\n\n\n\nbtw, drbinsomething's solution is equivalent to mine. he directly test P as a random variable while I tested mean nP as the random variable. He didn't mention it, but he took P to be asymptotically Normal as I did. '1.96' only applies when the distribution is Gaussian/Normal.
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# A statistical measure used in computing the correlation coefficient between two variables; the covariance is the mean of (x- x(bar))(yy(bar)) over all pairs of values for the variables x and y, where x(bar) is the mean of the x values and y(bar) is the mean of the y values. A statistical value measuring the simultaneous deviations of x and y variables from their means.\nhttp://helios.bto.ed.ac.uk/bto/glossary/c2.htm\n\n# A measure of the relation between two variables. The correlation coefficient is equal to the covariance of x and y divided by the product of the standard deviation of x and the standard deviation of y.\nhttp://www.indiainfoline.com/bisc/accc.html\n\nfor the formula, you can refer to this site - \nhttp://en.wikipedia.org/wiki/Covariance
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A statistic used as a measure of the dispersion or variation in a distribution. It is equal to the square root of the arithmetic mean of the squares of the deviations from the arithmetic mean. Simply put, standard deviation measures how spread out the values in a data set are. More precisely, it is a measure of the average difference between the values of the data in the set. If the data points are all similar, then the standard deviation will be low (closer to zero). If the data points are highly variable, then the standard variation is high (further from zero). the greater the standard deviation, the higher the risk. it is always positive, and NEVER negative. the formula you use is....well i can't write it because i don't have the symbols on my keyboard, but i'll try:\n\n\nσ= √(Σ[D-E]^2P)\n\nwhere:\nσ=sigma=standard deviation\nΣ=capital sigma= the sum of\nD=outcome\nE=expected value\n^2= squared\nP= Probability\n√=square root\n\nhere is an example:\n\n(Assumptions)(Outcome probability)(Outcomes)\n(Pessimistic)(0.2)                     (300)\n(Successful)(0.6)                     (600)\n(Optimistic)(0.2)                     (900)\n\n(sorry, the table won't keep the spaces, so i put brackets in)\n\nok, you need the expected value, which is the sum of all of the outcomes, times their probabilities, like so:\nexpected value = 300×0.2+600×0.6+900×0.2 = 600\n\n\nthen standard deviation would be calculated like so:\nstandard deviation = {(300-600)2×0.2+(600-600)2×0.6+(900-600)2×0.2}½ = 190\n\nhope that helped! oh, and if you don't understand, the "to the power of 1/2" part is the same as taking the square root.
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The central limit theorem still applies.\n\nWhen variables are correlated a principal component analysis will find the independent variables for which the CLT applies.
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A variogram is the basic tool in geostatistics for describing spatial correlation. Autovariogram is used to describe autocorrelation of a single continuous variable or a single class, and cross-variogram is used to describe cross-correlation between continuous veriables or classes.
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basically, you generate a list of random numbers (more about random number generation below), then use those numbers to calculate some output function.\nThen, you can have an "empirical distribution" about your output function.\nExample:\nyou can compute the present value of a financial project, GIVEN the values of, say, interest rates , sales level and inflation.\nYou also know that these three variables distribute normally and are correlated.\nThen you can generate a random scenario (one set os observations for each variable) and compute present value for it. You repeat the process a lot of times.\nSince you have a PV for each scenario you can observe its probability distribution.\nThat's it about Montecarlo Simulation.\nAbout Random number generation:\nif you know its cummulative probability distribution function (F(X)), then you can generate uniformly distributed random number in then (0,1) interval; let that be P.\nThe F-1(P) distributes with PDF F,where F-1 is teh inverse function of the cummulative function.\nThis method can be easily expanded to multivariate simulation using independt uniform variables, and the cholesky decomposition of the covariance matrix.\nNow, a note of care:\n\nMontecarlo methods can give the sensation that you can observe the distribution of a variable even if you know nothing about it. But, at the end, it's nothing more than a METHOD for computing PDF's  that are not easily expressed as functions.\n\nFinally, a note about computer power.\nMontecarlo methods are power hungry.Thus, using them to simulate PDF's of variables can indeed be expressed analytically is a complete waste of computin power and human time.
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i'm not sure exactly what this means.. usually you want to graph the distribution of some characteristic.  for instance, say you have a group of 1000 people, and you want to show how their heights are distributed.\n\noften the best way to do this is to use a histogram, which shows how many people are a particular height.  often (in the case of height, intelligence, etc) this leads to a bell-shaped distribution.\n\nhere is some more information about histograms:\nhttp://deming.eng.clemson.edu/pub/tutorials/qctools/histm.htm\nhttp://www.anu.edu.au/nceph/surfstat/surfstat-home/1-1-3.html\n\ncheers!
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Binomial Distribution, \nPoisson Distribution, \nNormal Distribution\n\nhttp://www.futureaccountant.com/probability/
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It is basically the behavior of the values in a population.  A distribution describes what the possible values of the population are, and which values are more common than others.
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"In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is defined as the square root of the variance.\n\nThe standard deviation is measured in the same units as the values of the population. For a population of distances in meters, the standard deviation is also measured in meters, whereas the variance is measured in square meters."\n\n\nIn essence, standard deviation is used for categorizing groups.\nFor example, the average IQ is 100, the standard deviation to the left would be 85, and deviation to the right would be 115. Two left standard deviations would be 60, two right standard deviations would be 130.\nSame applies to SAT scores, statistics, polls, etc.
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Find yourself a z-statistic table and look up critical value in the table.  Check your stats book on how to interpret the table.\nhttp://www.math2.org/math/stat/distributions/z-dist.htm
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The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields. It is a family of distributions of the same general form, differing in their location and scale parameters: the mean ("average") and standard deviation ("variability"), respectively. The standard normal distribution is the normal distribution with a mean of zero and a standard deviation of one (the green curves in the plots to the right). It is often called the bell curve because the graph of its probability density resembles a bell.