Normally distributed gaussian random variables

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Normally distributed (Gaussian) random variables possess a number of important properties that simplify their mathematical manipulation. One of these is that the mean vector and covariance matrix completely specify the joint probability distribution. This follows from the definition given above. In general, all of the moments and not just the first two are necessary to specify a distribution. Another property is that if x 1 , x 2 , · · · , x n are uncorrelated, they are also independent. Independent random variables are always uncorrelated, but the reverse need not always be true. A third property of jointly Gaussian random variables is that the marginal densities p x i ( x i ) and the conditional densities p x i x j ( x i ,x j | x k ,x l , · · · ,x p ) are also Gaussian. A fourth property is that linear combinations of jointly Gaussian variables are also jointly Gaussian. As a conse- quence, the properties of the original variables specify the properties of any other random variable created by a linear operator acting on x . If y = Ax , then E( y ) = A μ Cov( y ) = ACA t Using these properties, it is straightforward to generate vectors of random variables with arbitrary means and covari- ances beginning with random variables that are MVN. Central limit theorem The central role of MVN random variables in a great many important processes in nature is borne out by the central limit theorem, which governs random variables that hinge on a large number of individual, independent events. Un- derlying noise, for example, is a random variable that is the sum of a vast number of other random variables, each associated with some minuscule, random event. Signals can reflect MVN random variables as well, for example sig- nals from radar pulses scattered from innumerable, random fluctuations in the index of refraction of a disturbed air volume. 22
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Let x i , i = 1 ,n be independent, identically distributed random variables with common, finite μ and σ . Let z n = n i =1 x i The central limit theorem states that, in the limit of n going to infinity, z n approaches the MVN distribution. Notice that this result does not depend on the particular PDF of x i . Regardless of the physical mechanism at work and the statistical distribution of the underlying random variables, the overall random variable will be normally distributed. This accounts for the widespread appearance of MVN statistics in different applications in science and engineering. Rather than proving the central limit theorem, we will just argue its plausibility. Suppose a random variable is the sum of two other random variables, z = x + y . We can define a CDF for z in terms of a JDF for x and y as F z ( a ) = P ( z a ) = P ( x ≤ ∞ ,y a x ) = integraldisplay −∞ integraldisplay a x −∞ f xy ( x,y ) dxdy = integraldisplay −∞ dx integraldisplay a x −∞ f xy ( x,y ) dy The corresponding PDF for z is found through differentiation of the CDF: f z ( z ) = dF z ( z ) dz = integraldisplay −∞ f xy ( x,z x ) dx Now, if x and y are independent, f xy ( x,z x ) = f x ( x ) f y ( z
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