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Handbook of Probability by Ciprian A. Tudor, Ionut Florescu

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Chapter Ten

Gaussian Random Vectors

10.1 Introduction/Purpose of the Chapter

Gaussian random variables and Gaussian random vectors (vectors whose components are jointly Gaussian, as defined in this chapter) play a central role in modeling real-life processes. Part of the reason for this is that noise like quantities encountered in many practical applications are reasonably modeled as Gaussian. Another reason is that Gaussian random variables and vectors turn out to be remarkably easy to work with (after an initial period of learning their features). Jointly Gaussian random variables are completely described by their means and covariances, which is part of the simplicity of working with them. Estimating these joint Gaussians means approximating only their means and covariances.

A third reason why Gaussian random variables and vectors are so important is that, in many cases, the performance measures we get for estimation and detection problems for the Gaussian case often bounds the performance for other random variables with the same means and covariances. For example, the minimum mean square estimator for Gaussian problems is the same as the linear least squares estimator for other problems with the same mean and covariance and, furthermore, has the same mean square performance. We will also find that this estimator has a simple expression as a linear function of the observations. Finally, we will find that the minimum mean square estimator for non-Gaussian problems always has ...

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